In the bowl of a stand mixer, combine flour, salt, and yeast with the paddle attachment on low. Drizzle in the oil and warm water until the dry ingredients are moistened.
Switch to the dough hook and knead on medium, adding additional bread flour until the dough forms a ball. Knead for 3-5 minutes, adding more flour as needed to make a fairly stiff dough. Remove from the mixer bowl and transfer to a greased bowl. Cover with plastic wrap, and let rise until doubled in size (1-2 hours).
Dump the flatbread dough onto a lightly floured surface and divide it in half. If you plan on freezing half of it, now is the time. With the remaining half of the dough, divide it into four equal portions.
Stretch each piece into a rustic flatbread and transfer to a cornmeal-coated baking sheet.
Turn your grill to high and place the dough directly onto the grill.
Close the lid and cook until the bottoms are turning brown, 1-3 minutes. Move them around if necessary to ensure even cooking. Take the crusts off the grill and put them back on the baking sheet, grilled side up. I tried to put the ingredients directly onto the flatbreads and nearly burned my hand off. Spread the grilled side with sauce, cheese, toppings, and sprinkle with Parmesan cheese.
Return to the grill, close the lid, and cook an additional 2-4 minutes, or until turning brown.
Notes
To store Grilled Flatbread Pizza leftovers, wrap them tightly in aluminum foil or place them in an airtight container and refrigerate for up to 3 days. Avoid wrapping the pizza dough in plastic wrap directly, as it may expand and create a mess. Instead, store it in a lightly greased bowl covered with a damp cloth or lid.Reheat them in a preheated oven at 375°F (190°C) for 5-7 minutes, or until heated through. For a crispier crust, reheat on a grill or in a skillet.