Melt 1 tablespoon of the butter in a large, heavy skillet over medium heat; add shallots and cook, stirring often, until softened and lightly browned, about 5 minutes; add garlic and cook, stirring, 1 minute.
Meanwhile, mix beef, milk, salt and pepper in a medium bowl; stir in shallots and garlic; mix well; lightly form mixture into four patties.
Heat remaining tablespoon of the butter in the same skillet over medium-high heat; add patties; cook 5 minutes until browned on one side; turn; cook to desired doneness, about 5 minutes for medium; remove patties to a platter; keep warm; pour off any grease.
Add barbecue sauce and mustard to hot skillet; cook, stirring up browned bits, 2 minutes.
Place hamburger patties on bun bottoms; drizzle each with some of the sauce and place top bun on top.