Preheat your barbecue or indoor grill to medium-high. Preheat the oven to 250 degrees F. Line a baking sheet with parchment paper.
Brush the eggplant slices with the 2 Tbsp. oil; season with salt and pepper. Grill eggplant 2 minutes per side, or until nicely coloured and just tender. Set the eggplant on the baking sheet. Top each slice of eggplant with a slice of cheese and place the pan in the oven.
Spread the cut side of each bun with mayonnaise and pesto. Divide and top bottom buns with arugula. Set two slices of cheese-topped eggplant on each bottom bun, now set on 1 to 2 slices of tomato. Drizzle the tomatoes, eggplant and cheese with a little balsamic vinegar. Set on top buns and serve immediately.
Note: Mozzarella di bufala is sold in tubs in our deli department. If you cannot find it, you may use sliced bocconcini cheese in this recipe. Baby arugula is sold in plastic containers in our produce department.