Make the barbecue sauce: Heat the vegetable oil in a medium saucepan over medium heat. Add the onion and garlic and cook, stirring, until softened, about 4 minutes. Add the chili powder and cook, stirring, until toasted, about 1 minute. Stir in the ketchup, preserves, mustard, vinegar, soy sauce, Worcestershire sauce and 1/2 cup water. Bring to a simmer and cook until slightly thickened, about 10 minutes. Transfer to a blender and puree until smooth; let cool to room temperature.
Make the burgers: Combine the ground beef and 3 tablespoons of the prepared barbecue sauce in a bowl; mix with your hands until just combined. Form the meat into six 1/2-inch-thick patties; season both sides with salt and pepper.
Preheat a grill to medium. Toss the onion with 1 tablespoon of the barbecue sauce and the vegetable oil. Grill (use a grill basket if you have one), turning occasionally, until lightly charred, about 8 minutes.
Transfer 1/2 cup of the remaining barbecue sauce to a separate bowl for brushing. Increase the grill temperature to medium high and grill the patties, brushing with the barbecue sauce and flipping occasionally, about 8 minutes for medium rare; top with the cheese during the last minute and cover to melt. Toast the buns on the grill. Serve the burgers on the buns; top with the grilled onions, avocado slices and more barbecue sauce.