Heat a large skillet or griddle to medium high heat. Coat with oil. Add your onions and the hatch chile and season with your Salt and Pepper + Three Chiles. Cook until the onions are softened, for about 5 minutes. Remove from the heat and set aside.
Make your soaking mixture by adding the eggs, the full can of Pabst Blue Ribbon and season with Salt and Pepper + Three Chiles in a big rectangular container. Mix well until evenly combined. You can use an immersion blender to make it super smooth if desired.
Set your skillet or griddle to medium low heat. Add butter. Soak your slices of bread in your soaking mixture and place in the skillet or griddle. Cook for a couple of minutes and flip the slices of bread. Add 2 slices of cheese to each slice of bread and cook for a few more minutes until the cheese begins to melt. Add your hatch chile and onions to half of your bread slices and close your sandwiches. Cook for 6 more minutes, flipping your sandwiches every 2 minutes. Remove from the heat and serve while hot.