Remove the pizza dough from the refrigerator. Clear a 2-foot-square area on the counter that you can use later for rolling out the dough.
Place a large, heavy-bottomed skillet over medium-high heat. When the pan gets hot, add the olive oil, spreading it so it evenly coats the bottom of the pan. Add the mushrooms and cook them, stirring regularly, until they soften, about 7 minutes. Add the roasted peppers and garlic and cook for 1 minute more. Season with salt and pepper to taste and transfer to a medium bowl. Set aside (or if you want, you can let cool, cover, and refrigerate for up to 24 hours!)
Set the mushroom mixture, the pizza sauce, the grated cheese, and the pepperoni on a table near the grill. Prepare enough coals for a hot charcoal fire, or preheat your gas grill on high for 10 minutes with the lid closed.
While the grill is getting hot, generously dust a metal pizza tray or the back of a sheet pan with cornmeal. Also generously dust your clear counter area with flour.
Using a rolling pin and working from the center, roll out one ball of the dough into a 12-inch circle.
Transfer the dough to a pizza tray or the back of the sheet pan, making sure there is enough cornmeal to allow it to slide easily. Repeat with the other 3 pieces of dough, placing each one on a separate tray.
When the coals are ready or the gas grill is hot, bring one of the pizza trays to the grill and slide the dough onto the center of the grill grate. Grill for 2 to 3 minutes, or until golden brown. Turn over and immediately spoon a layer of sauce over the crust. Add some of the cheese, then some of the mushroom mixture, then some of the pepperoni.
Continue grilling for 2 to 3 more minutes, until the crust is cooked through (this may happen before the cheese melts completely, but that's okay).
Bring the pizza tray close to the grill and slide the pizza on to it. Transfer to a large cutting board. Have one of your kids use a pizza wheel to cut it into 6 pieces, and then transfer the slices to a large plate while you start working on the next pizza.