In large bowl, beat egg with water; stir in bread crumbs, onion, Dijon mustard, thyme, salt and pepper. Add beef; mix just until combined.
Divide into quarters; shape into balls. With finger, make well in each ball; place one-quarter of the cheese in well. Press meat over to enclose cheese. Press into four 3/4-inch (2 cm) thick patties. (Make-ahead: Layer between waxed paper in airtight container and refrigerate for up to 24 hours or freeze for up to 1 month. Thaw in refrigerator.)
Place patties on greased grill over medium heat; close lid and grill, turning once, until digital thermometer inserted sideways into centre reads 160°F (71°C), 12 to 14 minutes. Serve on buns.
Additional information : Variations Pork and Feta-Stuffed Burgers: Replace beef with pork; thyme with oregano; and Cheddar with crumbled feta. Top with sliced tomato and cucumber.
Chicken and Havarti-Stuffed Burgers: Replace beef with chicken or turkey, cooking to internal temperature of 175°F (80°C). Replace thyme with Italian herb seasoning; and Cheddar with tomato-basil Havarti cheese. Top with sliced tomato and fresh basil leaves.