FeChef, in it's simplest terms, a quesadilla is an extremely thin crust. Your quesadilla experiment proved, beyond any shadow of a doubt, what I've been forcefully asserting for quite some time- that cheese will never rise to it's fullest potential- it's rich, oozing, greasy gloriousness, without a thin enough crust. This is why authentic NY style pizza will ALWAYS beat the pants off of both inauthentic NY style (.1 TF) and American Style pies. The cheese on these thicker styles NEVER melts as well as it does on thinner crusts. In order for cheese to melt properly, it has to bubble and boil, and the only way it can bubble and boil is if it gets sufficient heat from below. If there's too much dough beneath it, the heat from the stone is handicapped from reaching the cheese during the bake time. No bubbling = no flavor.
I'm not sure what style of pizza you're currently making, but, if you want to recreate that quesadilla melt magic on a pizza, you should be making NY style with a thickness factor of no greater than .075. Anything else is a tremendous compromise from a perspective of legitimate mozzarella due diligence.
Thickness factor, is, by a very large margin, the most critical component for properly melted cheese. I would guess that it's about 85% of the cheese melt equation. A thin enough skin will go a very long way towards the perfect melt. The other 15% of the equation is worth talking about, though, for purposes of fine tuning.Heat Balance
Second to thickness is heat balance. While having proper top and bottom heat is vital for baking the crust, it's also important to the cheese. The heat from below causes the cheese to bubble and oil off, but the heat from above produces color. You don't get beautiful golden tan cheese without heat from both the top and the bottom.
In a home oven, this top heat means either a broiler, or, if the oven has no broiler in the main chamber- complex and difficult broilerless setups. I've talked about broilerless configurations elsewhere. If you have a broiler, though, you should be using it. In my experience, I've never seen an oven that allowed broiling during a clean cycle. This is one of the many reasons why I'm not a fan of cleaning cycles- it's pretty much impossible to get balanced top and bottom heat under these circumstances.
4 minutes is, imo, the perfect bake time, and, in order to hit that with your current stone, 650 appears to be the number. Without the cleaning cycle, if you still want to hit that 4 minute bake time, though, it's going to be time for a new hearth material. If your oven can do 550, then steel is the answer. Steel will give you 4 minute bakes @ 550 while still allowing the use of the broiler while the pizza bakes. Your nickname already takes you 3/4 of the way there. Instead of FeChef, you need to become Fe3C Chef
That's three iron atoms, and one carbon Stretching Technique
The thinness of the crust needs to be consistent in all areas of the pizza. All non rim areas of the skin should be the same height. To achieve this, edge stretching is critical. Home pizzamakers seem to have a hard time edge stretching. Many go straight from the finger poke to knuckle stretching. This creates a bowling effect to the skin. Not only will ingredients have a tendency to pool towards the center of the pie, but the thicker areas of the crust towards the rim will prevent the cheese from bubbling. This why you see countless pizzas on this forum with cheese in the center of the pizza with a great melt, but cheese towards the rim being a bit pale.Cheese selection
Based upon your quesadilla success, it sound likes you're working with a quality cheese. The thinness of a tortilla, though, is a best case scenario. In theory, you could have a cheese that melts beautifully on a quesadilla, but that doesn't melt quite as well on a slightly thicker NY style pie. I'm not saying that's happening in your case, but it doesn't hurt to be mindful of which cheeses melt well and which cheeses don't. You can follow every step that I've mentioned here and, if the cheese is bunk, it will melt poorly. Some defective cheeses brown too easily and some have a propensity for curdling. Ideally, one wants a good brand of commercial mozzarella and not the supermarket stuff.Cheese Grate
For NY style, always grate your cheese, never slice it or cube it, since the shape of the cheese impacts the manner in which it melts (larger pieces melt slower).Cheese/Crust Timing Synchronization
You might have the inclination to skip any potential alterations to your thickness factor and go straight to the list below. As I said before, don't. This is only for fine tuning. None of these tweaks will ever be able to compensate for the cheese impairing effects of a thicker crust.1) Cheese/Sauce Quantity
If you're not already weighing your cheese and sauce, then start doing so. Minor fluctuations in cheese and sauce quantity can completely change the nature in which the cheese melts and the timeframe when it's done baking. I've seen fluctuations as little as .5 oz. in either direction wreak havoc on a pizza. Sauce and cheese quantities are very subjective, so there are no 'perfect' amounts, but, from a cheese melt perspective, I find it essential to zero in to your ideal cheese/sauce quantities as quickly as possible and stick to them.2) Cheese/Sauce/Dough Temp
In a commercial setting, cheese and sauce temps can't really be altered, but, at home, if, say, your crust is finishes a little bit before the cheese is done, you can, on your next bake, let the cheese warm up at room temperature for a while. You can also control the temp of the sauce in the same way. Warmer sauce = faster cheese bake. If you're using refrigerated sauce, you can try making the sauce from room temp tomatoes. This is a bit extreme, but, as long as you don't cook the sauce, you can also nuke it very gently and warm it a bit.
Your dough temp should really be optimized for the best crust features, but, if need be, you can tweak it a bit so the dough bakes up a bit slower/faster. Cooler dough bakes up slower, warmer, faster.3) Sauce Consistency
This can be difficult to do, since tomatoes vary in consistency, but you want to, much like sauce/cheese quantity, dial in a standard consistency and stick with it, as water fluctuations dramatically change how the cheese reacts.4) Dough Formula
Again, much like your dough temp should be optimized for the best crust results, your formula should be completely crust-focused as well. It doesn't hurt, though, to be aware of all the factors that promote faster browning:
Protein in the flour
Enzymes in the flour (malted flour vs. unmalted)
More Sugar in the formula
Less yeast/Longer Fermentation time (to a point)
Summing up, thinner crust, thinner crust, thinner crust
Your quesadilla experiment and ensuing observations are a major game changer. I can't tell you how much time I spend trying to gently coax people into using smaller dough balls and/or stretching them further. Observing the superior cheese melt on a quesadilla might seem like a simple concept, but your experiment will be exhibit A in all of my future thickness factor related conversations. Thank you.