Wash and dry all produce. Adjust rack to top position and preheat oven to 425 degrees. Cut sweet potatoes into 1/2-inch-thick wedges. Drain pineapple over a small bowl, reserving juice. Thinly slice jalapeño into rounds, removing seeds and ribs for less heat.

Toss sweet potatoes on a baking sheet with a large drizzle of oil, salt, and pepper. Roast in oven, flipping halfway through, until browned and tender, 20-25 minutes. Meanwhile, add jalapeño to bowl with pineapple juice. Let marinate, stirring occasionally, until ready to serve.

Heat a drizzle of oil in a large pan over medium-high heat. Add pineapple and cook, tossing, until browned, 3-4 minutes. Remove from pan and set aside. Wipe out pan.

Divide pork into two equal-sized pieces, then flatten each into a 1/2-inch-thick patty. Place half the Monterey Jack in the center of each patty. Gently fold meat around cheese, shaping and sealing to create a cheese-stuffed patty. Season all over with salt and pepper.

Heat a large drizzle of oil in pan used for pineapple over medium-high heat. Add stuffed patties and cook to desired doneness, 4-6 minutes per side. While patties cook, halve buns. Place on baking sheet with sweet potatoes and toast in oven until browned, 2-3 minutes.

Fill buns with patties, barbecue sauce, pineapple, and as much marinated jalapeño as you like. Divide burgers and sweet potatoes between plates. Serve with remaining barbecue sauce on the side for dipping.