Heat 2 Tbsp. oil in a medium skillet over medium-high. Cook apple (which lends sweet and sour flavors to the chutney) and onion, stirring occasionally, until soft and lightly browned, 6–8 minutes; season with salt and pepper. Stir in sambal oelek and chili powder and cook until fragrant, about 1 minute. Add tomatoes and cook, breaking up with a spoon, until beginning to darken and juices have evaporated, about 4 minutes. Add cherries and ½ cup water, season with salt and pepper, and cook, stirring, until thick and jammy, 5–7 minutes. Transfer to a serving bowl; set aside.
Meanwhile, spread mayonnaise on 1 side of each slice of bread. Turn 4 slices mayonnaise side down, then evenly spread about ⅓ cup pimiento cheese on plain side. Top with 4 remaining bread slices, mayonnaise side up.
Brush a medium skillet with oil and heat over medium. Cook 2 sandwiches until golden brown on both sides, about 3 minutes per side. Brush skillet with oil and repeat with remaining sandwiches.
Cut sandwiches in half and serve with chutney alongside for dippage purposes.
Do Ahead: Chutney can be made 1 week ahead; cover and chill, or freeze 1 month.
2 tablespoons olive oil, plus more for brushing
1 Granny Smith apple, peeled, cored, cut into ¼-inch pieces
¼ medium red onion, chopped
1 teaspoon sambal oelek
½ teaspoon ancho chili powder
3 whole peeled tomatoes (from one 28-ounce can)
⅓ cup dried sour cherries
¼ cup mayonnaise
8 slices whole-wheat or white bread
1⅓ cups pimiento cheese spread, divided