2 tablespoons olive oil, divided
1/2 medium onion, diced
2 cloves garlic, minced
2 (14-ounce) cans white beans, drained and rinsed
1/2 cup breadcrumbs
1/3 cup shredded Parmesan cheese
1/4 cup fresh parsley leaves, coarsely chopped
1 large egg, beaten
3 tablespoons lemon juice
1 tablespoon anchovy paste
2 teaspoons Worcestershire sauce
2 teaspoons Dijon mustard
1/4 teaspoon kosher salt
1/4 teaspoon freshly ground black pepper
4 hamburger buns
Heat 1 tablespoon of the oil in a small skillet over medium heat until shimmering. Add the onion, and cook, stirring occasionally, until softened, about 5 minutes. Add the garlic and cook for 1 minute more. Remove the pan from the heat.
Mash the beans in a large bowl with a fork, keeping them a little chunky, or pulse in a food processor just until slightly broken down. Add the cooked onion, breadcrumbs, Parmesan, parsley, egg, lemon juice, anchovy paste, Worcestershire sauce, mustard, salt, and pepper to the mashed beans and mix until well combined. Cover the bowl and refrigerate for 1 to 2 hours.
Divide the mixture into 4 portions and shape each one into a 1/2-inch-thick patty.
Heat the remaining 1 tablespoon oil in a large frying pan over medium heat until shimmering. Without crowding the pan (and working in batches if needed), add the patties. Cook until the bottoms are golden-brown, 5 to 6 minutes. Flip the burgers with a flat spatula, and cook the other side for 5 to 6 minutes more.
Place the patties on the bottom halves of the buns, add your favorite toppings, and serve immediately.