Prepare a medium-high gas or charcoal grill fire. In a small bowl, combine the oil, mint, shallot, and vinegar. In another small bowl, combine the olives and walnuts. Beat the vinaigrette with a fork to blend, and then mix 2 tsp. into the olive mixture.
Arrange the eggplant and bread on a rimmed baking sheet. Brush the eggplant on both sides with olive oil and sprinkle with salt and pepper. Brush the bread slices on one side with oil, and sprinkle with salt and pepper.
Grill the eggplant, covered, until tender, about 3 minutes per side. Transfer to a plate. Grill the bread, covered, until toasted, 1 to 2 minutes per side.
Arrange the grilled bread on a cutting board, oiled side up. Top each slice with some of the mixed greens. Arrange the eggplant rounds atop the greens. Beat the vinaigrette with a fork to blend, and then spoon about 1 tsp. over each round. Slice the burrata about 1/2 inch thick and arrange over the eggplant. Sprinkle lightly with salt and spoon about 1 tsp. of the olive relish over each burrata slice. Cut the sandwiches in half and transfer to plates. Garnish with small mint leaves and serve.