Place flour, eggs, and breadcrumbs in 3 separate wide, shallow bowls. Working with 1 eggplant slice at a time, dredge in flour, then dip in egg and then breadcrumbs. Transfer to a wire rack set in a rimmed baking sheet and refrigerate 30 minutes.

Meanwhile, halve tomatoes and toss with olive oil and torn basil. Season with salt and pepper.

Heat about 1/4 inch vegetable oil in a large skillet over medium-high heat until shimmering. Add enough eggplant slices to fill but not crowd skillet and fry, flipping once, until deep golden brown and crunchy, 1 to 2 minutes on each side. (Lower heat slightly, if necessary.) Remove with a slotted spoon, drain on a clean wire rack set in a rimmed baking sheet, and season with salt. Discard used oil and repeat process with fresh oil and remaining eggplant.

Arrange eggplant and tomato mixture on a platter. Cut burrata in half, place next to tomatoes, and season with salt and pepper. Scatter remaining basil on top.