Preheat oven to 375 degrees F. Place beets in aluminum foil and drizzle with the 1/2 tablespoon of olive oil, wrap and place on a baking sheet. Bake until tender, about 1 hour 20 minutes. Allow to cool to room temperature and remove skins. Slice into thin rounds. Set aside.
Slice the goat cheese into 8 equal rounds. Place the flour, egg and panko on separate plates. Toss each slice of goat cheese in flour, then egg and finally in panko. Set aside. Heat the 1 cup vegetable oil in a medium skillet over medium-high heat. Fry the cheese until golden about 1 minute, flip over and repeat. Drain on paper towels. Cheese should not be runny.
Combine lettuces in a large bowl. Add grapes, croutons, 2 tablespoons olive oil and balsamic vinegar. Toss to coat. Add the sliced beets and sprinkle with salt and pepper to taste. Serve immediately.