Makes 4 sandwiches
Preheat the oven to 450 degrees with a rack positioned at the very top.
Place 3/4 cup olive oil, 3/4 cup extra-virgin olive oil, chile peppers, red pepper, garlic, and anchovies in a blender and blend until smooth. Transfer to a small bowl and stir in capers and mint. Set aside until ready to use.
Generously season eggplant slices with salt on both sides and arrange on a cooling rack over a baking sheet for 1/2 hour to drain; rinse and dry well with paper towels. Generously brush both sides of eggplant with olive oil. Arrange in a single layer on a clean baking sheet; season both sides with salt and black pepper. Transfer to oven and cook until nicely browned, turning eggplant over and rotating baking sheet midway through cooking, 10 to 12 minutes total.
Using the same pastry brush, stir olive oil mixture to combine well and brush some over eggplant on one side, making sure to coat well. Carefully transfer eggplant to a small mixing bowl in layers (use a spatula; the eggplant will be delicate), brushed side down. Brush other side before adding another layer and repeat until all eggplant slices have been brushed on both sides. Cover with plastic wrap and let marinate for 2 to 4 hours at room temperature. Occasionally spoon some marinade gathered at bottom of bowl over top of eggplant.
Arrange bread slices in pairs, in a single layer, on a work station. On one piece of each pair of slices, arrange basil leaves in a single layer to cover whole piece of bread (3 to 4 leaves). Remove eggplant from marinade, allowing any excess oil to drip back into bowl. Divide eggplant among the 4 sandwiches on top of the basil (spoon and drain some solids from marinade over eggplant). Season with a couple pinches of salt and pepper, to taste. Divide mozzarella slices among the sandwiches in an even layer. Place remaining slices of bread over mozzarella and lightly brush just the top bread with remaining marinade.
Heat a panini maker or Emeril XL grill to medium high or heat a grill pan over medium-high heat.
Place sandwiches in preheated panini maker (if using a grill or grill pan, place sandwich press or heavy plate on top of sandwich), in batches if necessary, and cook until bread is crisp, cheese has melted, and sandwich is heated through, 5 to 6 minutes. Remove from panini maker and cut in half. Repeat with remaining sandwiches and serve immediately.