Mix together green onions, mayonnaise, and salsa in a small bowl until well combined; spread over each bread slice. Layer turkey and pepper Jack cheese on 4 bread slices. Top with remaining bread slices to make 4 sandwiches.
Melt 2 tablespoons butter in a large skillet over medium heat. Fry sandwiches in hot butter until bottoms are lightly toasted. Add remaining butter to the skillet, then flip sandwiches over. Cook until browned on the other side and cheese is melted.