Any tomato can star in a sandwich— if you know how to play to its strengths.
Stir cream of tartar, salt, sugar, and pepper together in small bowl. Sprinkle mixture on both sides of tomato slices. Spread 1 teaspoon mayonnaise on 1 side of each slice of bread.
Place 2 slices of bread, mayonnaise side down, in 12-inch skillet. Cook over medium heat, moving bread if necessary for even browning, until underside is golden brown, 2 to 3 minutes. Transfer to cutting board griddled side up. Cook remaining 2 bread slices until underside is golden brown, about 30 seconds. Let cool until griddled side is crisp, about 2 minutes.
Evenly spread 2 teaspoons mayonnaise onto ungriddled sides of 2 slices of bread. Top with tomato slices in single layer, cutting tomato slices to fit if necessary. Place remaining bread slices on top of tomatoes, griddled side up. Serve.