Heat a panini press. Butter each piece of bread on one side, reserving 4 tablespoons of the butter. Lay 4 pieces of the bread, buttered-side down, on the panini press. Top with another piece of bread, buttered-side up. Press the bread until golden and slightly flattened, 2 to 3 minutes. Remove and repeat two more times until all the bread is toasted on one side. Set aside.
Mix together the eggs, half-and-half, 1 teaspoon salt and 1/2 teaspoon black pepper. Set aside.
Melt 2 tablespoons butter in a large nonstick skillet over medium-high heat. Spread the ham in the pan and sear until caramelized in spots, flipping once, 2 to 3 minutes total. Remove to a sheet pan and allow to cool.
Lower the heat to medium and melt the remaining 2 tablespoons butter in the skillet. Scramble the egg mixture until just set, 5 to 7 minutes. Remove the eggs to a sheet pan and spread them out to cool.
Meanwhile, mix together the mayonnaise, lime juice, 1/2 teaspoon kosher salt and some hot sauce to taste in a small bowl.
When the all the components have cooled, assemble the sandwiches by spreading all the pieces of the toasted bread with 1 tablespoon of the mayo mix. Next, lay a whole green chile on top of the mayo. Evenly distribute the shaved ham and scrambled eggs on top of the chiles. Top each with two slices of cheese and then the remaining slice of bread.
Wrap each sandwich individually in foil and place into 2 large resealable plastic bags. Place in the freezer and store up to 1 month.
To cook: Preheat the oven to 350 degrees F. Remove the sandwiches from the freezer and bake until heated through and the cheese has melted, 20 to 25 minutes.