Trim the tip off of the onions, then slice in half. Remove the skins and slice into thin half-moons. Heat the butter and the oil in a large skillet over medium heat, then add the onions. Stir to coat the onions with the butter mixture. Cook, stirring every 5 to 10 minutes, until the onions are caramelized and sweet. This will take anywhere from 40 minutes to an hour, depending on the size of the onion slices and the type of onions. When they have reached your desired level of caramelization, stir in the chicken broth, making sure to scrape up any browned bits from the bottom of the pan. Season to taste with salt. Set aside.
Meanwhile, combine the ground beef and ground pork in a bowl and mix just until combined. Divide the mixture into 4 equal pieces and form into patties slightly larger than the rolls. Season both sides with salt and pepper. Refrigerate until needed.
Preheat the oven to 400ºF/240ºC. Place the bacon in a single layer on a foil lined baking sheet. Bake the bacon until crisp, 15 to 20 minutes. Keep an eye on it to make sure it doesn’t get too crispy. Remove to a paper towel lined plate.
Heat the Big Green Egg to high heat, about 500ºF/260ºC. Place the burger patties (straight from the fridge) onto the EGG and cook with the dome closed for 3 minutes. Flip the burgers and cook an additional 3 minutes. Close all of the vents and let the burgers sit in the EGG for another 5 minutes. Add 2 pieces of cheese to the top of each patty, close the dome, and let the patties sit another minute to melt the cheese.
To assemble the burgers, put one patty on the bottom half of each of the Cobblestone Bread Co.™ Toasted Onion Rolls. Top each patty with 1 oz of ham, then 2 slices of bacon. Place ¼ of the caramelized onions on top. Spread mayonnaise on the top half of the bun, then place on top of the burger. Serve immediately.
Makes 4 sandwiches