How to Make It
Preheat the broiler and position a rack 8 inches from the heat. In a bowl, combine the oil and garlic. Brush the eggplant slices with 1 1/2 tablespoons of the garlic oil, arrange on a broiler pan and season lightly with salt and pepper. Broil the eggplant for about 8 minutes, turning once, until tender and deep golden. Transfer to a platter to cool. Repeat the process with the summer squash, using 1/2 tablespoon of the garlic oil and broiling for about 6 minutes.
In a medium bowl, toss the shiitakes and red pepper with the remaining 1 tablespoon of garlic oil and season with salt and pepper. Arrange the pepper, skin side up, on the broiler pan and add the shiitakes. Broil for about 6 minutes, turning the shiitakes once, until the mushrooms are tender and golden and the pepper skin is charred. Add the shiitakes to the platter. Transfer the pepper to a small bowl, cover with plastic wrap and let steam for 15 minutes. Peel the pepper and cut it into thin strips.
Line a large sturdy baking sheet with parchment paper. Cut the puff-pastry sheet in half to form two 10-by-4-inch rectangles. On a lightly floured surface, roll the pastry to form two 14-by-5-inch rectangles, about 1/8 inch thick. Set the pastry rectangles on the prepared baking sheet and refrigerate them until firm.
Preheat the oven to 375°. Arrange the cheese on the pastry, leaving a 1-inch border. Arrange the eggplants, shiitakes and squash on the cheese and crisscross the pepper strips on top. Fold the pastry sides up and pinch the corners. Refrigerate until the pastry is firm, about 10 minutes.
Bake the tarts for about 40 minutes, or until they are puffed and golden and the bottoms are cooked. Let cool slightly on the baking sheet, then transfer to a rack. Cut the tarts crosswise into 2-inch slices and serve warm or at room temperature.