In a large cast-iron skillet over medium-high heat, add the bacon and onion. Cook until the bacon is crisp and the onion is softened, 6 to 8 minutes. Remove to a paper towel-lined plate. Add the sausage to the same skillet and, using the back of a wooden spoon, break it up into small bits. Cook until golden brown, 5 to 7 minutes. Add the bacon and onion back to the skillet, along with the ham. Pour in the whisked eggs and use a rubber spatula to scramble the eggs and meat. Cook until the eggs are set, about 5 minutes, then add the cheese, salsa verde and chiles and stir everything together. Season with salt and pepper.
Divide the scramble among the 12 tortillas. Roll up the tortillas, leaving the ends open. Wrap each burrito in foil. Serve immediately.
To make ahead: Refrigerate the burritos after assembling; reheat the next day at 350 degrees F for 35 minutes in a pan covered with foil.