For the chiles: Heat the broiler. Place the New Mexico chiles, jalapenos and bell peppers cut-side down on a foil-lined broiler pan or baking sheet. Broil until the skins are charred and blackened, turning if needed, about 8 minutes. Transfer to a bowl and cover with plastic wrap to steam, about 10 minutes. When cool enough to handle, peel the charred skin carefully (or under running water) and chop into very small pieces. Mix together with the cilantro, vinegar and garlic in a bowl. Season with salt and black pepper and set aside. This step can be done in advance.
For the sandwiches: Turn the oven to 350 degrees F. Rub the ham steak with the cut side of the garlic and place in a baking dish. Add 1/4 cup water, cover with foil and bake until warmed through, about 10 minutes. Remove and cut into 4 even pieces, cutting around the bone and discarding if needed.
Place 2 slices of provolone on one side of each roll. Toast the rolls on a baking sheet in the oven until lightly golden brown and the cheese has melted, about 4 minutes.
Divide the ham steaks between the cheesy sides of the rolls.
Melt 1 1/2 tablespoons of the butter in a large nonstick skillet over medium heat until bubbling. Crack 4 eggs at a time into the skillet, sprinkle with salt and black pepper and fry sunny-side up until the eggs are set, about 5 minutes. Remove from the pan and place 2 eggs each onto 2 sandwiches. Wipe out the pan and repeat with the remaining butter and eggs.
Scoop 2 tablespoons of the roasted green chiles over the eggs in each sandwich. Close the bun and enjoy!