Preheat oven to 350°F and line a baking sheet with parchment paper. Spread beans evenly on the baking sheet and bake for 12-15 minutes until slightly dried out, mixing halfway through. Transfer beans to a large bowl and mash partially with a potato masher, leaving some whole beans.
Place salsa in a fine-mesh sieve and stir a few times to drain excess liquid. Transfer the salsa to the bowl with the black beans. Add the breadcrumbs, onions, eggs, mustard, cumin, salt, and pepper. Place in fridge for 10 minutes to set.
Form the mixture into patties with about 1/3 cup mixture in each. This should make about 6-8 patties.
Cook on Stovetop: Heat half the oil in a large nonstick grill pan or skillet over medium heat. Add 4 patties and cook for 4-5 minutes per side until browned. Repeat with the remaining oil and patties.
Cook on Grill: Brush the grill surface with a little oil. Cook patties over medium-high heat for 4-5 minutes per side until browned.
Bake in Oven: Place patties on a lined baking sheet and bake at 375°F for 20 minutes, flipping halfway through.
Serve the burgers on buns with your favorite toppings.