Combine the jack cheese and chopped bacon in a bowl. Put the beef in another bowl and season with salt and pepper; use a rubber spatula to mix it in. Flatten 1/4 of the meat mixture in the palm of your hand. Squeeze together about a tablespoon of the cheese and bacon mixture to make a little nugget; put it in the center of the ground beef. Bring up the sides of the patty over the filling, making sure the cheese is completely covered by the meat. Flatten slightly for a nice patty shape. Set them side by side on a platter in the refrigerator while preparing the grill.
Place a large grill pan on 2 burners over medium-high heat or preheat an outdoor gas or charcoal barbecue and get it very hot. Brush the grates with oil to keep the burgers from sticking. Grill the burgers for 8 minutes per side for medium. (The USDA recommends cooking ground beef until it is no longer pink and the internal temperature is 160 degrees F.) Remove the burgers to a clean side plate while you toast the buns.
Rub the grill rack with more oil and place the buns cut-side down, toast for 1 minute. Serve the burgers with your favorite condiment and any garnish you like, such as lettuce, sliced tomato, onion, and/or avocado.