Step 1: Line a rimmed baking sheet with foil and spray with cooking spray. Divide the sausage into 4 equal pieces (1/2 pound each). Wet your hands with cold water and form the meat into 4 balls. Arrange several inches apart on the baking sheet.
Step 2: Spray the bottom and lower third of the beer or soda can with cooking spray. (Don’t skip this step.) Firmly press the bottom of the beer can into each meatball, forming a cup; the bottom and sides of the cup should be of fairly uniform thickness. Wrap a slice of bacon around the outside of each cup. Carefully remove the beer can. Using your fingers, fuse any cracks in the bottom or sides of the burgers. Refrigerate, uncovered, for 30 minutes to 1 hour.
Step 3: Season the inside of each burger with salt and pepper. Secure the bacon with toothpicks.
Step 4: When ready to cook, set up your grill for indirect grilling and preheat to medium-low, 300 degrees. Brush and oil the grill grate. Carefully transfer the burgers, open side up, to the grill grate away from direct heat. Toss the wood chips on the coals. Cover the grill. If using a gas grill, place all of the wood chips in the smoker box or in a smoker pouch and run the grill on high until you see smoke, then reduce the heat to 300 degrees.
Step 5: Grill the burgers for 30 minutes. Using paper towels, blot any excess fat pooled in the depression made by the beer can. Sprinkle 1 ounce of grated cheese in the cup of each burger. Break an egg over the cheese, keeping the yolk intact if desired. (You can also beat the eggs lightly before adding them to the burgers.)
Step 6: Continue to cook the burgers for 20 minutes more, then top the burgers with the remainder of the cheese.
Step 7: Cook for 10 more minutes, or until the meat reaches an internal temperature of 160 degrees and the eggs are set to your liking. (Generally, the yolks will be fully cooked by the time the whites firm up.) Serve on a grilled bagel, English muffin, or with toast points.
Because of the depth of the burgers, the egg yolks will be cooked (not runny) by the time the whites set. You can incorporate the yolks by beating the eggs first (as if for scrambled eggs), or if you insist on runny yolks, you can reserve the yolks and add them once the whites are beginning to firm up.