Cook the onions in 1 tablespoon butter with 1/4 teaspoon salt over medium-low heat for 35 to 40 minutes until caramelized.
Shape two broad thin patties, season with remaining salt and pepper and cook over medium-high heat for 2 to 3 minutes per side to 160°F.
Toast one face of each bread slice lightly in 1 tablespoon butter.
Layer pepper Jack, patty, onions, drained chiles and more cheese against the toasted inner faces and butter the outsides.
Griddle over medium-low heat until crisp and melted, 3 to 4 minutes per side.