How to Make It
In a medium saucepan, heat 2 teaspoons of olive oil. Add the apple and apricots and cook over moderate heat, stirring, until just softened, 5 minutes. Add the sugar, vinegar, cinnamon stick and orange zest and juice. Cook over moderately low heat, stirring occasionally, until thickened, 8 minutes. Remove from the heat; discard the cinnamon stick. Stir in the mustard and season with salt and pepper. Let the mostarda cool completely.
Preheat a griddle over moderately low heat. Brush one side of each bread slice with olive oil. On the other side of 4 bread slices, spread with the mostarda (1/4 cup each), then top each with 3 slices of cheddar. Close the sandwiches, set them on the hot griddle and top with a large baking sheet weighted down with cans. Cook over moderately low heat until toasted, 3 minutes per side. Open the sandwiches and tuck 1/2 cup of the arugula into each. Close the sandwiches, cut in half and serve hot.