Preheat oven to 375°F. Place an oven-safe rack over a large rimmed baking sheet covered with foil. Lightly spray rack with cooking spray.
Season sliced eggplant evenly with 1/2 tsp. salt and 1/4 tsp. black pepper. Combine breadcrumbs and Parmesan in a wide-rimmed bowl. Place egg and 1 Tbsp. water in a separate bowl; whisk to combine.
Dunk each slice of eggplant in egg mixture followed by breadcrumb mixture, pressing to adhere. Place eggplant slices on prepared baking rack. Lightly coat eggplant slices with cooking spray. Bake at 375°F for 30 to 35 minutes, rotating pan once halfway through.
Heat olive oil in a large skillet over medium. Add shallots and garlic; cook 2 minutes, stirring occasionally, until softened. Add tomatoes, vinegar, soy sauce, sugar, and remaining 1/2 tsp. salt and 1/4 tsp. black pepper. Simmer over medium heat until mixture reduces and thickens, about 8 to 10 minutes. Remove from heat and stir in 1 tsp. of the tarragon.
Combine mayonnaise, lemon juice, and remaining 2 tsp. tarragon in a small bowl; mix to combine.
Slice loaf of bread (or baguette) in half, lengthwise. Arrange mozzarella slices on one half, and pop bread under the broiler for 2 minutes, until the cheese is melted and bread is golden.Arrange eggplant slices over the melted cheese, followed by the tomato relish.On the other half of bread, evenly spread tarragon mayonnaise and top with fresh arugula. Combine top half with bottom half of sandwich and slice into five.