When it is too cold outside to grill, this is the perfect way to have a nice, juicy burger. Two cups may seem like a lot of shallots, but they really shrink down when you cook them.
Heat a nonstick skillet over medium-low heat. Add 2 teaspoons oil; swirl to coat.
Add shallots and ¼ 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" thick patty. Press a nickel-sized indentation in the center of each patty. Sprinkle evenly with the remaining ¼ 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 ¼ cup arugula on bottom half of each bun; top with 1 patty and ¼ of shallots.
Spread about 2 teaspoons mayonnaise on top half of each bun; place on top of burgers.