Pickled chicories amp up this tuna melt, a lunchtime classic, and are a great use for leftover mixed chicories. The pickles can add a pop of tangy sweetness to salads or be served as a condiment with grilled or roasted meats.
Cut chicory leaves crosswise into 1/2-inch pieces; discard woody stems. Place chicory leaves and shallot in a lidded glass jar or ceramic container; set aside. Combine 3 cups water, rice vinegar, red wine vinegar, sugar, salt, ginger, peppercorns, and crushed red pepper in a medium saucepan. Bring to a simmer over medium-high, stirring occasionally to dissolve sugar and salt. Remove from heat; let cool to room temperature, about 1 hour. Pour mixture through a fine wire-mesh strainer into a bowl; discard solids. Pour strained liquid over chicory mixture in jar; cover with lid, and seal. Let stand at room temperature about 1 hour. Chill at least 8 hours or overnight or up to 1 week.
Stir together 1/3 cup chopped pickled chicories, 1 teaspoon brine from pickled chicories, tuna, mayonnaise, dill, salt, and pepper in a medium bowl until combined.
Preheat oven to broil with rack 6 inches from heat source. Brush 1 side of each bread slice with oil; arrange bread slices in an even layer, oiled sides up, on a rimmed baking sheet. Broil in preheated oven until toasted, about 3 minutes. Remove from oven; flip bread over. Top each slice with 2 cheese slices. Return to oven; broil until cheese is melted and beginning to brown, 2 to 3 minutes. Remove from oven.
Divide tuna salad evenly between 2 toast slices, spreading to edges. Top evenly with 1 tablespoon chopped pickled chicories, and cover with remaining 2 toast slices. Serve immediately.
Pickled chicories can be made up to 1 week ahead and stored in an airtight container in refrigerator.