Dice eggplant into 1/2-in. cubes and put in a colander. Toss with 1/2 tsp. salt and let drain 1 hour.
Meanwhile, make seasoned bread crumbs: Heat panko in a medium (not nonstick) pan over medium heat, stirring occasionally, until golden brown, 2 to 5 minutes; stir in thyme and a couple of pinches each salt and black pepper. Let cool.
Bring a medium pot of salted water to a boil and fill a bowl with ice and cold water. Using a sharp paring knife, core tomatoes by cutting a cone around stems. Score bottom of each tomato with a small X.
Blanch tomatoes 20 seconds, immersing two at a time. With a slotted spoon, transfer to ice water. When cool, lift tomatoes to a plate and peel them. Slice tomatoes thickly and arrange over two-thirds of a platter.
Make vinaigrette: In a small bowl, combine shallot, red wine vinegar, 1/2 tsp. salt, and 1/4 tsp. white pepper. Whisk in 1/4 cup oil.
Pour oil for frying into a small pot to a depth of 1 1/2 in. Insert a deep-fry thermometer and heat over medium-high heat until oil registers 325°.
While oil is heating, rinse eggplant and pat dry with paper towels. Set a metal cooling rack in a rimmed baking pan and put next to stove.
Put flour in a shallow bowl, egg in another, and seasoned panko in a third. Coat eggplant cubes in flour. Using a slotted spoon, lift out cubes while shaking off excess flour; drop cubes into egg. Using a second slotted spoon, lift out cubes and drop into bread crumbs; toss to coat. Now use the first (dry) spoon to transfer eggplant to hot oil.
Fry eggplant in two batches if necessary until crisp and golden brown, 5 to 7 minutes per batch, and transfer with the dry slotted spoon to drain on rack in pan. Season croutons with salt.
Drizzle tomatoes with 3 to 4 tbsp. vinaigrette. Slice cheese thickly and arrange over tomatoes; sprinkle with salt and pepper to taste.
In a medium bowl, toss arugula with 1 1/2 to 2 tbsp. vinaigrette. Arrange arugula on platter next to tomatoes. Scatter with warm eggplant croutons and torn basil leaves, if using.