Season both sides of the green tomato slices with salt and set aside; this will draw out some of the moisture.
Toss the whole canned tomatoes with the vinegar, brown sugar, oil and a large pinch of salt. Place the whole tomatoes on a parchment-lined baking sheet. Bake until slightly caramelized and dried out a bit, about 20 minutes. Set aside.
Set up a standard breading station with three shallow bowls, one with the flour, one with eggs and one with panko. Season the green tomato slices with black pepper. Dredge each tomato slice in the flour, shake off the excess, dip in the egg, let excess drip back into the bowl, and then coat in panko. Set aside on wire rack and repeat with the remaining slices.
Heat 1/2 inch of oil in each of 2 large straight-sided saute pans over medium heat. Working in batches, fry a few of the tomato slices at a time, turning them once, until golden brown, 4 to 5 minutes a side. Using tongs, transfer to a paper towel-lined baking sheet. Repeat with the remaining tomato slices. Lower the oven temperature to 175 degrees F. Transfer the tomatoes to a wire rack set over a sheet pan. Place in the oven to keep warm.
Butter the insides of the buns and toast until golden and crispy. Lay the Burrata on the bottom of the buns, then top with hot green tomatoes, 4 to 5 roasted tomatoes, a spoon of hot giardiniera and some basil. Close buns, cut each in half and serve with applause.