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, salt, pepper, and half the BBQ Seasoning (you’ll use the rest later). Roast on top rack, flipping halfway through, until browned and tender, 20-25 minutes.

Meanwhile, 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.

In a large bowl, combine beef, remaining BBQ Seasoning, and a big pinch of salt and pepper. Divide mixture in half (divide into quarters for 4 servings). Flatten each piece into a wide, roughly ½-inch-thick circle. Set aside half the pepper jack. Divide remaining cheese between the center of each circle. Fold edges of meat around pepper jack, 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 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 extra), then top with reserved pepper jack; cover pan until cheese melts.

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. Serve with potatoes and any remaining veggies on the side.