Make the herb butter: Mix the butter, parsley, garlic, lemon juice, Worcestershire sauce, 1/2 teaspoon salt and a few grinds of pepper in a small bowl. Transfer to a piece of plastic wrap and form into a 4-inch log. Wrap and freeze until firm enough to slice, about 20 minutes. Slice the butter log into 8 rounds.
Make the burgers: Preheat a grill to medium high. Divide the ground beef into 4 mounds, place a slice of butter on each, then gently shape the beef around the butter to enclose. Press the beef into four 4-inch-wide patties (about 1/2 inch thick). Refrigerate until ready to grill. Set aside the remaining butter for serving.
Season the burgers on both sides with salt and pepper. Grill until well marked, 3 minutes. Flip and top each with a slice of cheese. Cook 3 more minutes for medium rare, covering during the last 15 seconds to melt the cheese. Grill the buns until toasted.
Spread the remaining herb butter on the bottom buns. Serve the burgers on the buns with mustard, onion, tomatoes and lettuce. Skewer cornichons on toothpicks and insert into each burger.