Heat a nonstick skillet over medium-low heat. Add 2 teaspoons oil; swirl to coat. Add shallots and 1/4 teaspoon salt; cook 15 minutes or until golden brown, stirring occasionally. Stir in vinegar; cook 1 minute. Remove from heat; keep warm.
Gently combine garlic and beef. Divide the meat mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Press a nickel-sized indentation in the center of each patty. Sprinkle evenly with the remaining 1/4 teaspoon salt.
Heat a large cast-iron skillet over medium-high heat. Add the remaining 1 teaspoon oil to pan; swirl to coat. Add patties, and cook for 3 minutes on each side or until desired degree of doneness. Top each patty with 2 tablespoons cheese; cover and cook 1 minute or until cheese melts.
Place 1/4 cup arugula on bottom half of each bun; top with 1 patty and one-fourth of shallots. Spread about 2 teaspoons mayonnaise on top half of each bun; place on top of burgers.