Combine the first 6 ingredients in a bowl. Lightly moisten your hands with cold water. Form meat mixture into 4 equal balls. Flatten each ball into a thick patty. Set a square of cheese in the centre of each patty. Mold meat around cheese to enclose. Flatten each filled burger into a ¾" thick patty.
Preheat you grill to medium-high. Lightly oil the grill. Grill the burgers 4 minutes per side, or until entirely cooked through and the centre reaches 160°F or above on an instant-read meat thermometre. Sandwich the burgers in the buns with your choice of garnishes.