Slice tomatoes and mozzarella about 1/4-inch thick. Rinse and pat basil dry (you can keep it whole or tear into smaller pieces). Set aside.
Preheat a large skillet to medium heat and add 1 Tbsp. olive oil. Prepare eggplant slices by coating lightly in flour, then egg (shaking off excess), and then panko, all in separate dishes. Once skillet is hot, add eggplant slices and fry, in batches, about 3 minutes per side until golden brown. Remove to a paper-towel lined plate.
Combine dressing ingredients in a small bowl and whisk until well-combined. Taste and adjust seasonings as needed.
On serving plates, build stacks by alternating tomato, mozzarella, eggplant, and basil. Repeat until you have 2 equal-sized stacks. Drizzle with dressing.