Adjust rack to top position and preheat oven to 425 degrees. Wash and dry all produce. Cut potatoes into ½-inch-thick wedges. Halve, peel, and thinly slice onion. Halve, core, and thinly slice green pepper into strips. In a small bowl, combine ketchup with hot sauce to taste.

Toss potatoes on a baking sheet with a large drizzle of oil, half the BBQ Seasoning (you’ll use the rest later), salt, and pepper. Roast on top rack, flipping halfway through, until browned and tender, 20-25 minutes.

While potatoes roast, heat a large drizzle of oil in a large pan over medium-high heat. Add onion and green pepper. 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.

Meanwhile, in a large bowl, combine beef, remaining BBQ Seasoning, and a big pinch of salt and pepper. Form beef into two wide, roughly ½-inch-thick rounds (four rounds for 4 servings). Set aside half the pepper jack. Divide remaining cheese between centers of each round. Fold edges of meat around cheese, shaping and sealing to create cheese-stuffed patties, each slightly wider than a burger bun. Season all over with salt and pepper.

Heat a large drizzle of oil in pan used for veggies over medium-high heat. Add patties and cook to desired doneness, 3-5 minutes per side. In the last 1-2 minutes of cooking, top each patty with a layer of veggies (you may have some left over), then top with reserved pepper jack; cover pan to melt cheese.

While patties cook, halve and toast buns. Spread cut sides of buns with spicy ketchup. Place smothered patties on bottom buns; drizzle with any remaining hot sauce if desired, then add top buns. Divide burgers between plates and serve with potatoes and any remaining veggies on the side.