Blot the chopped green chiles with paper towels, then stir them with the cumin and salt.
Divide half the Monterey Jack over 4 bread slices and spread the chile mixture evenly over the cheese.
Add the remaining cheese and close with the other bread slices.
Butter the sandwich exteriors and place them in a skillet over medium-low heat.
Cover and cook for 3 to 4 minutes per side, then uncover and cook each side for 1 minute more until crisp and fully melted.
Rest the grilled cheese sandwiches for 1 minute before cutting diagonally.
