Place a pizza stone in the oven and preheat to 500 degrees. Let stone heat for 30 minutes.
Heat olive oil in a skillet over medium heat. Add onion slices and sauté until softened. Add grape tomato halves and cook until charred. Remove from heat.
Cut dough in half and working with one-half at a time, roll out into a 10-inch circle on a piece of parchment paper on a solid surface like a pizza peel, large cutting board, or baking sheet.
Spread half the pesto over the surface of the pizza leaving a 1-inch border around the crust. Take half the burrata and break into pieces and distribute over pesto. Then place half the onion and tomato mixture over the cheese. Sprinkle half the pine nuts over the top.
Slide pizza and parchment paper onto pizza stone. Bake for 7 minutes. Check pizza, rotate if necessary and continue baking until the crust is dark golden brown, typically 10-15 minutes total depending on the thickness of the dough.
Garnish with basil, salt, pepper, and crushed red pepper flakes if desired. Repeat with the rest of the ingredients.