How to Make It
Preheat the oven to 400°. Lightly brush a large rimmed baking sheet with olive oil. Arrange the eggplants on the baking sheet, cut side down, and bake until just tender, about 20 minutes. Let cool slightly, then cut the eggplant halves lengthwise into thirds. Discard the eggplant skin and the very seedy portions, then cut the eggplants into 1/2-inch chunks.
Preheat the broiler. Lightly brush the red peppers with olive oil and set on a heavy baking sheet. Broil the red peppers 6 inches from the heat, turning the peppers occasionally, until lightly charred all over, about 15 minutes. Transfer to a bowl, cover with plastic wrap and let steam for 10 minutes. Peel, core and seed the peppers, then cut them into 1/4-inch dice.
In a large saucepan, heat the 2 tablespoons of olive oil until shimmering. Add the onions and cook over moderately low heat, stirring occasionally, until softened, about 7 minutes. Gently stir in the eggplants and peppers and season with salt and cayenne. Add the stock and bring to a boil. Simmer over moderate heat until the flavors blend, about 10 minutes.
Meanwhile, in a mortar, pound the garlic with 1/2 teaspoon of salt until smooth. Stir in the mayonnaise, lemon juice and a pinch of cayenne.
Stir the 2 tablespoons of mint into the soup and ladle into shallow bowls. Top with a dollop of aioli and a sprinkling of mint and serve.