Preheat the oven to 375 degrees F. Line a baking sheet with a silicone liner and another baking sheet with a wire rack.
For the eggplant: In a bowl, toss the tomatoes with the balsamic vinegar, olive oil, brown sugar and some salt. Evenly place the tomatoes on the lined baking sheet. Place in the oven and roast until slightly caramelized and dried out a bit, 20 to 30 minutes. Set aside.
Set the eggplant slices on the wire rack (if you don't have a rack you can use a colander set in the sink). Sprinkle salt on each side of the eggplant and let the salt do the work for 15 to 30 minutes.
Once the salt has released moisture from the eggplant, pat the slices dry with paper towels. Then sprinkle each side of the eggplant with pepper.
Set up a standard breading station with the flour in one shallow dish, the eggs in another and the panko mixed with the Parmesan in a third dish. Dredge each eggplant slice in the flour, shake off any excess, dip in the egg, let any excess drip off and then coat in the panko. Set aside on the wire rack and repeat.
In a large cast-iron skillet, heat 1-inch vegetable oil to 350 degrees F.
Working in small batches, fry the eggplant slices, turning once, until golden brown, 3 to 5 minutes a side. Using a pair of tongs, transfer back to the wire rack. Repeat with the remaining eggplant. Make sure to season the eggplant as it comes out of the oil.
For the sandwich build: Butter and griddle both sides of the buns until golden and crispy (you can toast the buns as well). Tear the burrata over the bottom buns, then add 2 hot eggplant slices, some roasted tomatoes, a few pickled cherry peppers and some fresh basil leaves on each bun. Serve to applause.