Heat oil in a small saucepan over medium heat. Using one of the larger settings on a box grater, grate onion and transfer to pan. Cook 5 minutes, stirring occasionally.

Slice one of the peaches in half and remove core. Holding onto each half, grate flesh of peach on box grater until you reach the skin; discard skin. Add grated peach, garlic, ginger, and tomato paste to saucepan and cook 2 minutes, stirring occasionally. Add vinegar, maple syrup, soy sauce, salt, and chili flakes; bring mixture to a boil. Reduce to medium-low and simmer for 10 to 12 minutes, until thickened. Set aside.

Preheat oven to 500°F. Lightly grease a 12-inch cast-iron skillet (or, alternatively, sprinkle with corn meal). Stretch pizza dough to a 13-inch circle. Fit dough into the bottom and slightly up the sides of skillet. Place skillet over medium-high heat on stovetop; cook until bottom of crust is dry and beginning to brown, ~8 minutes. Remove from heat; prick crust several times with a fork.

Spread BBQ sauce over crust in skillet. Slice remaining peach into thin slices and scatter peach and red onion overtop. Brush exposed crust with olive oil and place skillet in preheated oven; bake until crust is golden-brown, about 10 to 12 minutes. Remove from oven and top with torn pieces of burrata, fresh basil, and/or arugula, and extra red chili flakes, if desired.