Combine 1 tablespoon olive oil and butter in a skillet over medium heat. Add onions to the pan, season with salt and pepper and cook 20 minutes to caramelize then stir in olives and anchovy paste and cook 2 minutes more. Remove from stove.
While the onions cook, combine chicken with 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 4 patties and cook 6 minutes on each side. Melt the Swiss cheese on each burger the last 2 minutes of cook time. Loosely tent the burgers under foil to melt the cheese. Serve the burgers on buns topped with lettuce and caramelized onion topping. Serve with potato chips alongside.
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