1. Preheat a grill for direct, medium-high grilling. If using charcoal, push 2/3 of hot coals to 1 side of grill and spread remaining 1/3 on other side to create a multi-zone fire. If using a gas grill, reduce heat on 1 side to medium.
2. In a medium nonstick skillet over medium-high heat, cook sausage, breaking up any large chunks with a wooden spoon, until lightly browned and cooked through, about 5 minutes. Remove from skillet with a slotted spoon and transfer to a small bowl, leaving fat behind in skillet. Add bell pepper and onion to skillet and cook, stirring occasionally, until soft, about 6 minutes, adding 1 Tbsp. olive oil to skillet if mixture looks dry. Transfer to a separate small bowl.
3. Dust a 14-inch-wide pizza peel with flour or cornmeal and generously flour a work surface. Roll dough into a 12-inch round, adding more flour as needed to prevent sticking. Transfer dough to pizza peel and brush with remaining 2 tsp. olive oil.
4. Have your toppings within reach of the grill. Slide dough onto hottest part of grill; cover grill. Cook, rotating as needed, until bottom is blistered and browned, about 2 minutes. Flip dough onto cooler side of grill, cooked side facing up.
5. Spread pizza sauce over crust and top with shredded mozzarella, leaving a 1/2-inch border. Spread pepper-onion mixture and sausage on top. Cover grill and cook until bottom crust has browned and cheese has melted, 2 to 3 minutes, occasionally uncovering and checking bottom of crust to ensure it's not burning. If it is, reduce heat (if using a gas grill) or move off the grill (if using charcoal). Transfer pizza to a cutting board and let stand 2 minutes before slicing and serving.