Place the drained and rinsed black beans in a large bowl and use a fork to coarsely mash them, leaving some whole beans for texture.
Add the lightly beaten egg, seasoned bread crumbs, 1/2 cup grated cheddar, smoked paprika, garlic powder, and a generous pinch of kosher salt and pepper to the bowl; mix until combined.
Use your hands to bring the mixture together; it should be moist and hold when pressed into a patty. If needed, mash more beans with the fork to achieve cohesion.
Divide the mixture into four equal portions and form each into a ball.
Heat 1 to 2 tablespoons avocado oil in a large cast-iron skillet or nonstick pan over medium heat.
Add one bean ball to the skillet and immediately press it flat with the back of a jar or flat glass (lightly sprayed with nonstick spray if desired) to form a very thin patty; use the glass to press edges together if pieces separate.
Cook the patty 1 to 2 minutes until the bottom is golden and crisp, then gently flip and cook the other side until brown, golden, and crisp, about another 1 to 2 minutes.
Place a slice of cheddar on each patty after flipping and let it melt while the second side finishes cooking.
Assemble burgers by placing patties on buns and topping with tomato slices and house sauce; serve immediately.