Mash green chiles with a fork; spread over 3 slices of bread. Top with cheese and remaining bread. In a shallow bowl, whisk eggs and milk until blended.
In a large skillet, heat 1 tablespoon butter over medium heat. Dip both sides of sandwich in egg mixture. Place in skillet; toast 2-3 minutes on each side or until golden brown and cheese is melted. Repeat with remaining butter and sandwiches. If desired, serve with salsa.