Preheat oven to 400°. Heat a large ovenproof frying pan over medium-low heat. Add 1 tbsp. oil and brush all over pan.
Press dough with your hands, working on a floured surface, into a 13-in. round. Press into frying pan so it comes up sides slightly. Cook until golden on bottom, 5 minutes. Slide onto a plate.
Add remaining oil to pan along with the vegetables, salt, and chile. Cook over medium-high heat, stirring, until peppers begin to brown. Transfer to a plate.
Invert dough back into pan, golden side up, and press down into pan. Sprinkle with 1 cup mozzarella, leaving a border around edge. Scatter vegetables over cheese and sprinkle with remaining mozzarella.
Bake pizza in pan until crust is crisp, about 15 minutes. Cut into wedges.