Heat 1 tbsp. oil in a frying pan. Add shallots and cook, stirring often, until softened and browned, about 4 minutes. Remove from pan and set aside.
Meanwhile, combine chicken, garlic, rosemary, salt, and pepper. Form mixture into 4 patties. Add remaining 1 tbsp. oil to same pan and cook patties, turning once, until cooked through, about 5 minutes.
Place a lettuce leaf on bottom half of each bun and top with a patty. Top patties with shallots. Mix mayonnaise with blue cheese, spread on bun tops, and set on patties.
Note: Nutritional analysis is per burger.