John, funny you mentioned that you like your cheese more done. We have some members that prefer the cheese to look one way and others another. When I first started making pizzas, I like the cheese spotted as you. Over time, I have come to prefer a cheese that is melted and just beginning to oil off, but I no longer desire a spotted cheese.
To get a spotted cheese using the LBE, simply lower the ceiling closer to the the pie or elevated the pie into your ceiling. In the weber charcoal grill specifically, you can use the ash catcher, drill a hole through the center of it and attach it to the lid. The dome of the ash catcher will force the heat closer to the pie and char your cheese.
Another method of achieving this is to use a stone in the lid instead of a metal disk. I believe members Ronzo, Buceriasdon, Essens1, and many others use a stone in the top lid. The stone absorbs heat and either reflects or radiates heat to help char the cheese. You can contact them via PM or if they read your post, they can link you to threads where they have posted their builds.
Weber charcoal grills can be found on Craigslist for as little as $10-$20. Buddy that with a $40 burner and a few small mods and you can have a hell of a pizza oven. My LBE makes fantastic pizzas and I'll continue to bake in it despite having a WFO now, especially during winter times or when I just want to bake 2 pies.
You'll have a lot of fun with the LBE, but it does require some tweaking to get optimal results. If you want to read about my successes and failures with the MBE (mini black egg) prior to building my LBE just search for "MBE" and you'll see it. Don't forget to take time to read the original LBE thread (60+ pages). That thread is full of ideas for getting the most out of your LBE.