Preheat the oven to 325 F (162 C). Drain and rinse the black beans in a colander, then shake off as much water as you can. Dump the black beans onto a rimmed baking sheet and bake in the preheated oven for 20 minutes (the beans should look almost dried out).
While the beans are in the oven, heat 1 Tablespoon of olive oil in a large nonstick skillet over medium heat. Add the finely diced green peppers and onions to the pan and saute until the peppers are softened and the onions are transparent (about 7 minutes). Add in the minced garlic and saute until aromatic (about 1 minute).
Add the peppers and onions from the pan to a large mixing bowl, then gently dab the sauteed vegetables with a paper towel to absorb some of the moisture. Add the feta cheese, eggs, BBQ sauce, mayonnaise, Worcestershire sauce, chili powder, cumin, paprika, and salt to the bowl. Mix until fully combined.
Once the beans are done baking, remove them from the oven and let them cool for about 5 minutes. Then use a potato masher to mash the beans on the baking sheet, while leaving some bigger chunks behind. (Mashing the beans on the baking sheet is a lot easier and more effective than trying to mash them in the bowl. The flat surface of the baking sheet is super helpful here.)
Before finishing the burgers, prepare the grill. Clean the grill plates (your grill needs to be clean in order for the burgers to not stick). Preheat the grill to 325 F (anywhere between 300-350 F will work).
Add the mashed beans and panko bread crumbs to the bowl and mix until fully incorporated.
Line a baking sheet with parchment paper. This is where you’ll form and transport your bean patties (lining it with parchment paper will stop the patties from sticking and make it easier to keep the patty shape when you transfer it to the grill).
Using a 1/3 measuring cup, scoop the bean mixture from the bowl. Place onto the prepared baking sheet. Gently form the patties to the size of your bun on the sheet, making them about 1/2 inch thick.
Once the grill is up to temp, oil the grill. Brush oil over the side of the patty that will be face down on the grill, then use a spatula to transfer the oiled patties to the oiled grill. Close the lid and cook until the bottom is lightly browned and has shrunk in a little bit (about 9 minutes). Do not move the patties before they’re ready to flip.
Before flipping, brush the top side of the burgers with olive oil, then carefully flip the burgers over. Close the lid and cook for about 7 minutes, then add sliced cheese on top and cook for 2 more minutes. (If you’re not adding cheese, cook the burgers for 9 minutes on the second side.)
Remove the patties from the grill and place onto your burger buns. Top with slices of tomatoes, red onion, and preferred condiments. Serve hot.