Clean and preheat the grill to 400 degrees F. This can also be done in the house over a burner.
Cook/grill the ham for about 3 minutes each side or until lightly charred. Remove from the heat and cut into bite-sized pieces. Set aside in a warm spot.
Melt 1 tablespoon of the butter in a 10-inch cast iron skillet over medium-low heat. Swirl the butter around to evenly coat the skillet.
Pour 2 eggs in the skillet. Cook for about 2 to 3 minutes or until they begin to set up around the edges. Sprinkle the eggs evenly with ½ cup Mozzarella cheese.
Place 1 tortilla on top of the eggs and cheese and mash down slightly to seal the ingredients together.
Cook for about 3 minute or until the eggs are cooked and easily release from the skillet.
Flip the tortilla over and sprinkle ½ cup of pepper jack cheese evenly on top. Sprinkle ½ of the green chiles followed by half of the chopped ham.
Fold half of the tortilla over to meet the other side and mash slightly.
Cook for about 3 to 4 minutes, each side, or until golden brown. Repeat with the remaining ingredients for the second quesadilla. Slice and serve immediately.