Roast the Hatch and jalapeño chiles under the broiler until blackened, about 5 minutes per side. Place the chiles in a paper sack or plastic food-storage bag, close it tight and let the chiles steam for 20 minutes. After the chiles have steamed, take them out of the bag and gently rub the chiles to remove the skin. Remove the stems and seeds from the chiles and then dice.
Preheat the oven to 375°F. In an ovenproof skillet, melt the butter on medium-low heat. Add the onion and cook until translucent, about 5 minutes. Add the garlic and cook for 30 more seconds. Remove the skillet from the heat and stir in the diced chiles. Evenly distribute on top the cream cheese and the shredded Muenster and Monterey Jack, then pour in the half-and-half. Bake uncovered until the cheese is bubbling, about 15-20 minutes. Remove from the oven, garnish with the cilantro and gently stir to combine everything. Add salt to taste.
Meanwhile, in a large skillet heat up the butter on medium heat. Add the onions and while occasionally stirring, cook until fragrant and almost translucent, about 2-3 minutes. Lightly salt and pepper the rib-eye slices and add them to the skillet. Cook the meat with the onions for 2-3 minutes, stirring occasionally, until the meat is just starting to brown.
To assemble the sandwiches, evenly divide the meat and the onions between the toasted and buttered buns and top the meat with the cheese sauce.