Grilled-Vegetable Agrodolce

Preheat grill for direct-heat cooking. Season zucchini, squashes, and eggplants with salt and pepper. Lightly drizzle with oil; grill until tender, about 3 to 4 minutes per side (if browning too quickly, move to cooler part of grill).

In a large skillet, heat oil over medium. Add onion and 2 tablespoons sugar. Season with salt and cook, stirring often, until lightly browned, 6 to 8 minutes. Remove from heat and stir in vinegar, remaining 1/4 cup sugar, and 1 cup water. Return to medium high heat and cook until reduced by half, 9 to 10 minutes. Stir in raisins, capers, and bay leaves; season with salt. Let cool slightly. Arrange grilled vegetables in a 9-by-13-inch baking dish or other shallow vessel and pour marinade mixture over top. Drizzle generously with oil. Let cool completely, cover with plastic, and refrigerate at least 12 hours and up to 2 days.

When ready to serve, toast bread on grill or under broiler, flipping once, until lightly charred, 2 to 3 minutes total. Drizzle with oil. Arrange marinated vegetables and cured meats on a platter. With kitchen shears, cut burrata into quarters; drizzle generously with oil. Season with salt and pepper. Layer toast with cheeses, and top with cured meat, marinated vegetables, and parsley.