Whisk the olive oil, balsamic vinegar, Italian seasoning, salt, and pepper together. Pour the whisked ingredients over the eggplant rounds and toss well to coat.
Slice the halloumi into ½-inch thick slices.
Heat the grill. Arrange the eggplant slices on the grill, and cook for about 6 minutes on each side, until the eggplant is cooked.
Grill the halloumi on both sides, until golden grill marks appear, about 2 minutes per side.
Assemble the burger. Spread mayonnaise on 2 halves of a burger bun. Take one half of a bun, and stack the eggplant, halloumi, and 1 teaspoon of pesto on to it. Top with tomato and lettuce. Place the other half of the bun on top. Serve immediately.