Step 1Line a rimmed sheet pan with a silicone mesh grill mat or aluminum foil and lightly oil it. Divide the sausage into 4 equal portions (½ pound each). Wet your hands with cold water and form the meat into 4 balls. Arrange the meat portions several inches apart on the sheet pan.
Step 2Oil the bottom and lower third of a beer or soda can. (Don’t skip this step.) Firmly press the bottom of the beer can into one of the sausage balls to form a cup. the side of the cup should come about 1 inch up the side of the can. Carefully twist out the beer can and repeat with the rest of the meat. Use your fingers to mold the bottoms and sides of the cups into uniform thickness and to fuse any cracks in the meat. Wrap 2 bacon strips around the outside of each cup and secure with toothpicks.
Step 3Refrigerate the meat cups, uncovered, for 1 hour. The burgers can be prepared to this stage up to 8 hours ahead.
Step 4When ready to cook, set up your grill for indirect grilling and heat to medium-high. Brush or scrape the grill grate clean and oil it well. If working on a silicone mesh grill mat, simply slide it onto the grate away from the fire. Otherwise, gently pry the burgers off the foil-lined sheet pan and transfer, open-side up, to the grate away from the fire. Add the wood chunks or chips to the coals if using charcoal, or place in the smoker box of your gas grill. Lower the grill lid.
Step 5Indirect-grill the burgers until browned and firm, 20 to 30 minutes. Using paper towels, blot any excess fat pooled in the depression made by the beer can. Crack an egg into the cup of each burger. (If this makes you nervous, crack the egg into a ramekin and carefully pour it into the cup.) Sprinkle salt, pepper, grated cheese and chopped green onion over each egg.
Step 6Lower the grill lid and continue indirect grilling the breakfast burgers until the sausage meat is browned, the cheese is melted and the eggs are cooked to taste, 8 to 12 minutes.
Step 7Transfer the breakfast burgers to a platter. Grill the English muffins, cut sides down, directly over the fire, 1 to 2 minutes. Place 1 breakfast burger on the bottom of each muffin (I like to serve the burgers open-face), and get ready for one of the most outrageous breakfasts of your life.