Sprinkle salt on both sides of the eggplant slices and arrange in a single layer on a triple layer of paper towels on a baking sheet. Place another triple layer of paper towels on top. If there are additional slices, arrange them on top of the paper towels in a single layer, then top them with another triple layer of towels. Let the salt do the work for 30 to 45 minutes.
Firmly press on the eggplant slices to remove as much liquid as possible, then wipe off any excess salt. Pepper both sides of the eggplant. Preheat the oven to 175 degrees F.
Set up a standard breading station with 3 pie plates or small baking dishes. Add some flour to one, the beaten eggs to another and the breadcrumbs stirred together with the Parmesan to the third. Season each station lightly with salt and pepper.
Dredge each eggplant slice first in the flour, then shake off the excess. Dip in the egg, letting any excess drip, and then coat in the breadcrumbs. Set aside on a wire rack and repeat with the remaining eggplant slices.
Heat 1/2 inch of canola oil in 2 large, straight-sided saute pans over medium heat until hot. Line a baking sheet with paper towels.
Working in small batches, fry a few eggplant slices in each pan, turning once, until golden brown, 4 to 5 minutes per side. Using tongs, transfer to the towel-lined baking sheet to drain. Fit another baking sheet with a wire rack to hold the eggplant warm in the oven while you finish frying the remaining slices.
To build the sandwich: slice the burrata into disks (try to keep its shape) and arrange in a single layer on the bottom portion of the loaf. Put a single layer of the warm, crispy eggplant on top, then top with the roasted red peppers and drizzle with the Balsamic Vinaigrette. Use toothpicks to secure the top, then slice and serve.
Put the vinegar in a small saucepan with the garlic and shallots and bring to a simmer over medium heat. Simmer, uncovered, until the vinegar reduces to about 1/2 cup. Let cool for about 5 minutes.
In a blender, pulse together the vinegar mixture, honey, Italian seasoning and some salt and pepper. While the machine is running, slowly drizzle in the olive oil until the dressing is combined and emulsified. Adjust the seasoning if necessary, or add more oil if you want it less acidic.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.