Combine 1 tablespoon olive oil and the butter in a skillet over medium heat. Add onions to the pan, season with salt and pepper and cook for 20 minutes to caramelize. Stir in the olives and anchovy paste and cook for 2 minutes more. Remove from the stove.
While the onions cook, combine the chicken with the herbes de Provence, fennel seed, lemon zest, garlic, salt and pepper.
Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Form four patties and cook for 6 minutes on each side. Melt the Swiss cheese on each burger during the last 2 minutes of cook time. Loosely tent the burgers under the foil to melt the cheese.
Serve the burgers on buns topped with lettuce and the caramelized onion topping. Serve with potato chips alongside.