In the bowl of a food processor, combine black beans, 1/2 red onion, the egg, panko breadcrumbs, Worcestershire, chili powder, cumin, paprika, oregano, garlic, and a pinch of salt and pepper. Processor until combined and only fine chunks remained. No need to fully puree. If the mixture seems too sticky, add flour 1 tbsp at a time. Set batter in the fridge for 10 minutes - this will help the ingredients soak in the moisture. Keep in mind they will still be a bit sticky.
In a clean food processor or blender, puree pineapple chunks. Scoop into a pan along with the remaining ingredients for the barbecue sauce. Bring to a boil over medium heat, then lower to a simmer and cook, stirring occasionally, for at least 15 minutes until thickened.
Heat 2-3 tbsp olive or avocado oil over medium heat. Form 4 burgers from the dough and set into the skillet. Cook 3-4 minutes, then flip and cook an additional 3-4 minutes until lightly browned. Set slices of pepperjack over each patty, place a lid over the pan, and cook until cheese is melted, about 2-3 minutes. Remove to a plate.
Add another drizzle of olive oil to the pan. Saute peppers and red onion with a pinch each of salt and pepper for 2-3 minutes, until just tender but still crunchy.
Toast burger buns in the broiler or over an open flame. Spoon barbecue sauce over bottom bun and top with greens of choice. Set a burger patty and a heap of fajita veggies overtop. Finish with more barbecue sauce and the top bun.