1: Prepare the pizza dough. Let it rise once, punch it down, form into two balls, and let it rise again until doubled in bulk.
2: Meanwhile, set up your grill for indirect grilling and heat to high. Heat a pizza stone as well. (It should be about 600 degrees.)
3: Stretch out half the pizza dough to form a 10- by 12-inch circle. Place it on a pizza peel generously dusted with cornmeal.
4: Sprinkle the pizza with half the gruyere. Arrange half the bacon, shiitakes, and scallions on top, leaving 3 3-inch circles for the eggs. Crack 3 eggs in the circles, taking care not to break the yolks. Sprinkle the pizza with the half the Parmigiano (try to avoid the eggs) and drizzle the whole shebang with olive oil. Sprinkle the pizza with salt, pepper, and hot red pepper flakes. Prepare the other pizza the same way.
5: Slide the pizza onto the hot pizza stone. Close the grill lid and grill until the bacon is sizzling and browned, the eggs are cooked, the cheese is melted, the crust is puffed and browned and the whole thing looks good enough to dive into. Cook the other pizza the same way. Cut into wedges and serve.