In bowl, combine ground beef, Dijon mustard, anchovy, tomato paste, salt and pepper. Mix just until combined. Form into four equal patties.
Place patties onto greased grill over medium-high heat. Cook, flipping once, until desired doneness. About ten minutes for well done.
Set oven to broil; arrange rack in middle. On parchment-lined baking sheet, arrange pre-sliced burger buns (tops and bottoms together) a few inches apart. Top each bun with 1/2 tablespoon of tomato sauce and one tablespoon cheddar, pushing fallen cheese toward edges of buns. Broil until cheese is melted and bubbling, about one to two minutes.