Divide beef into 4 equal portions and shape into 3-inch patties. Form cheese into two 1½ inch rounds; place cheese atop two patties. Top with another patty and tightly seal the edges. 

Prepare a grill to medium-high or preheat broiler. Brush grill with oil. Grill or broil bread slices until lightly toasted, about 1 minute per side. Brill or broil burgers, turning once, to preferred doneness, about 4 minutes per side for medium. Assemble burgers as desired