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Prepare the vegetables: Slice the eggplants crosswise into 12 rounds, each about 1/2-inch thick. (Discard the small ends.) Using a serrated knife, slice the tomatoes into 12 slices, each about 3/8-inch thick, discarding the small ends.
Pour 1/4 cup of the olive oil on a rimmed baking sheet. Lay the eggplant slices, cut sides down, in the oil and turn over so both sides are coated. Season with Traeger Veggie Shake or coarse salt and black pepper. Do the same with the tomato slices, adding more olive oil if necessary.
Lay the eggplant slices directly on the grill grate in a single layer. Grill until tender, about 20 minutes, turning once with a spatula. If you have room on your Traeger, put the pan with the tomato slices on the grill grate at the same time and grill for 15 minutes. Do not turn the tomatoes as you do not want them to break apart. (If you do not have room on your grate to grill both vegetables at once, do them in batches.) If desired, grill the vegetables up to 1 day ahead; cover and refrigerate.
Assemble the stacks: Lay an eggplant slice in each of 4 wells of a jumbo muffin tin. (Alternatively, stack the ingredients in clean tuna cans - which will also keep the stacks from collapsing - or directly on the baking sheet you used to roast the tomatoes.) Continue to build the stacks by alternating ingredients, finishing with a tomato slice. Sprinkle a tablespoon of Parmesan cheese on each stack.
Place the muffin tin on the grill grate and roast the stacks for about 10 minutes, or until the cheese begins to soften and melt.
To serve, transfer the stacks to plates. Drizzle with the remaining olive oil and a little balsamic vinegar. Garnish with a sprig of fresh basil. Can be served warm or cold.