Place torn bread into a bowl; sprinkle with milk and Worcestershire sauce. Mix in egg until combined; season with basil, rosemary, and black pepper. Add veal; mix until evenly combined. Divide and shape into four patties.

Preheat an outdoor grill for medium and lightly oil the grate.

Cook on the preheated grill until desired doneness, about 5 minutes per side for medium. Place 1 slice cheese on each burger; cook until melted, about 1 minute. Serve on buns.