OK, I took another shot at this last night, when a friend from KC, who is familiar with Pizza Shoppe, was visiting. I did my mix in a food processor as before, but this time tried processing just the oil into the dry ingredients first, before adding the water/yeast mixture.
I made a 1-pound batch (1 pound flour) for two 12-inch pizzas. 24-hour countertop rest in a sealed container. Split the dough into two pieces, then rolled each out, dusted with flour, folded in half, dusted with flour again, then folded that in half (to make 4 layers) then rolled that out to the final size, docked and put in the cutter pan and trimmed. Parbaked the crust on a stone at 500F for 4 minutes, let cool on a rack, then topped and baked at 450F on the stone for about 7 minutes.
The result was pretty good. Photos attached. I had tried doing a 16-layer version the last couple of weeks but it took so much rolling out that the layers kind of welded themselves together and you really didn't get a lot of separate layers like I wanted. As you can see in these pix, there are some nice separate layers. My friend said she thought it was better than Pizza Shoppe, but I still think something is missing.
I am using a fabulous smoked pepperoni from our local meat market which is just jaw-droppingly delicious, along with a couple variants of their Italian sausage. Dang, it's good.
I'll keep working on this -- it looks like we're going to KC sometime next month, at which point I'll definitely visit Pizza Shoppe to get a pizza and evaluate what they might be doing differently.