Preheat grill to 375 degrees.
Divide the ground beef into fourths. Roll 1/4 pound of meat into a ball then press the meat onto a sheet of waxed paper or another non-stick surface. Shape the flattened ground beef into a rectangle then sprinkle salt and pepper over the meat.
Place one cheese stick on the edge of the beef square then roll the meat over the cheese stick. Press the meat around all edges of the cheese until it is completely covered by ground beef.
Once the grill has reached a temperature of at least 375* degrees place the Cheese Stuffed Burger Dogs onto the hot grill. Close the lid and cook for 5 minutes before rotating the burger dogs to cook an additional 5 minutes. Turn them two more times to create grill marks on all sides and for a total cook time of 20 minutes.
After 20 minutes on the grill the cheese should begin to seep out of the ends of the burger dogs and the juices should be dripping clear. An additional minute or two may be needed. (Or for those who prefer a more rare hamburger a minute less on each grill rotation should do the trick.)
Remove the burger dogs from the grill and serve immediately with hog dog buns and your favorite condiments.