Mix the mayonnaise, salsa, and green onions in a small bowl. Spread the seasoned mayonnaise evenly on each slice of bread. Layer the turkey and cheese on 4 of the slices. Top with remaining bread to make 4 sandwiches.

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Melt 2 tablespoons butter in a large skillet over medium heat. Fry sandwiches in butter until lightly toasted. Add remaining butter to skillet, turn sandwiches over.Cook until the cheese is melted, and the bread is browned.