Puree 1 cup of beans with ¼ cup of sun-dried tomatoes, olive oil, tomato paste, and garlic powder in the small bowl of a food processor.
Put remaining beans in a medium bowl and mash them a little bit with a fork. Stir in the bean-tomato mixture, basil leaves, cheeses, breadcrumbs, and remaining tomatoes. Add salt and pepper to taste, then fold in the beaten egg.
Line a baking sheet with parchment paper. Divide the burger mixture into 4 mounds on the baking sheet, then shape each mound into a patty. Bake the burgers until they’re lightly browned on the outside and feel dry to the touch in the center, 20-25 minutes.
Serve the patties on buns, with your choice of toppings.