In a skillet, cook bacon over medium heat until crispy. Drain on paper towels. When cool enough to handle, crumble bacon. Drain all but 1 Tbsp. fat from skillet and sauté onion until tender, about 3 minutes. Add garlic and sauté 1 minute longer. Transfer to a plate and let cool completely.
In a large bowl, combine ground chuck, paprika, barbecue sauce and reserved bacon and onion. Season liberally with salt and pepper. Gently but thoroughly mix with fingers. Divide into 16 pieces. Form each piece into a 3-inch patty. Sprinkle cheese over 8 patties, cover each with one of remaining patties and press to seal.
Preheat gas grill to medium (or light a charcoal fire and let burn to a gray ash). Brush grill with oil. Grill hamburgers until desired doneness, 4 to 6 minutes per side for medium-rare. Remove patties and let stand for 2 to 3 minutes before serving on buns.