Pour the Balsamic vinegar into a sauce pot. Bring to a boil, and then reduce heat and simmer for 5 minutes. Let cool and place in sealed container. It should be the consistency of a thick syrup. If it gets too thick, it can be thinned with additional vinegar.
Dredge an eggplant slice in the flour, then dip it in the egg, and finally dredge it in the breadcrumbs. Shake off any excess breading and transfer the eggplant to a wire rack. Repeat with the remaining eggplant. Allow the eggplant to dry for 15 minutes once all of the slices are finished. Pour the oil to a depth of 1/4 inch in a small saute pan. Once the oil is warm, work in small batches frying the eggplant slices until golden brown, about 2 minutes per side. Using tongs, transfer to a paper towel-lined baking sheet and season with salt to taste. Repeat with the remaining eggplant. Once cool, cut the eggplant slices into thirds.
Place arugula and basil in a mixing bowl and toss with 2 T olive oil, garlic, salt and pepper.
Layer the greens, eggplant strips, tomatoes, and fresh mozzarella on dinner plates. Drizzle the Balsamic vinegar reduction over the top.