Adjust rack to top position and preheat oven to 450 degrees. Wash and dry all produce. Cut sweet potatoes into 1/2-inch-thick wedges (like steak fries). Halve, peel, and thinly slice onions. Core, deseed, and thinly slice green peppers. In a small bowl, combine ketchup and hot sauce (to taste).

Toss sweet potatoes on a baking sheet with a large drizzle of oil, salt, pepper, and half the BBQ Seasoning. Roast on top rack, flipping once, until browned and tender, 20-25 minutes.

Meanwhile, heat a large drizzle of oil in a large pan over medium-high heat. Add onions and green peppers and cook, stirring, until browned and softened, 7-9 minutes. Season with salt and pepper. Turn off heat; remove from pan and set aside. Wipe out pan.

In a large bowl, combine beef, remaining BBQ Seasoning, and a large pinch of salt and pepper. Divide mixture in fourths. Flatten each piece into a wide, roughly 1/2-inch-thick circle. Divide half the cheese (you’ll use the rest later) between the center of each circle. Fold edges of meat around cheese, shaping and sealing to create cheese-stuffed patties. Season all over with salt and pepper.

Heat a large drizzle of oil in pan used for onions and peppers over medium-high heat. Add patties and cook to desired doneness, 3-5 minutes per side. Carefully top each patty with a layer of onions and peppers (you may have extra), then top with remaining cheese. Cover pan until cheese melts, about 1 minute.

While burgers cook, halve buns and place on a second baking sheet. Toast in oven until golden brown, 2-3 minutes. Spread cut sides of buns with spicy ketchup. Top bun bottoms with smothered patties. Drizzle with any remaining hot sauce if desired. Serve with sweet potatoes (and any remaining onions and peppers) on the side.