In a large bowl, combine the rinsed and drained white beans, 1 tablespoon of the olive oil, and the minced garlic. Use a fork to lightly mash the beans and mix until the beans, oil, and garlic are blended but still slightly chunky.

Add the diced red bell pepper, bread crumbs, Worcestershire sauce, egg, and chopped chives to the bowl. Stir until evenly combined.

Crumble the ground turkey into the bowl. Season with kosher salt and freshly ground black pepper to taste. Gently mix with a fork or your hands just until the mixture is blended (do not overmix).

Fill a medium bowl with water and moisten your hands. Divide the mixture into four portions and shape each into a patty about 3 inches in diameter and 3/4 inch thick, dipping your hands in the water as needed to prevent sticking.

Heat the remaining 2 tablespoons of olive oil in a large nonstick skillet over medium heat until shimmering.

Add the patties to the skillet and cook until the bottoms are browned, about 4 to 5 minutes. Flip each patty and continue cooking until the burgers are firm to the touch and register 165°F on an instant-read thermometer, about 5 to 8 more minutes.

Transfer the cooked burgers to the toasted bread slices and serve immediately.