How to Make It
Preheat the oven to 400° and line 2 large rimmed baking sheets with foil. On opposite ends of 1 baking sheet, separately toss the eggplant with 3 tablespoons of the olive oil and the zucchini with 1 tablespoon of the olive oil. Season the eggplant and zucchini with salt and pepper and spread each vegetable in a single layer. On opposite ends of the second baking sheet, separately toss the beet and the sweet potato each with 1 tablespoon of the olive oil. Season with salt and pepper and spread in a single layer. Roast all of the vegetables until tender, about 10 minutes for the zucchini and about 25 minutes for the remaining vegetables; rotate the baking sheets halfway through baking. Let the vegetables cool slightly. Keep the oven on.
Meanwhile, in a medium saucepan, heat the remaining 2 tablespoons of olive oil. Add the carrots, onion, apricots and garlic and cook over moderate heat, stirring occasionally, until just softened, about 5 minutes. Add the wine and 3/4 cup of water and simmer over moderately low heat until the carrots are very tender, 15 minutes. Remove from the heat and stir in the vinegar, fish sauce, Worcestershire, soy sauce, ground ginger and fennel. Scrape into a blender and let cool slightly. Puree until very smooth, adding water 1 tablespoon at a time to loosen the sauce, if necessary. Season the carrot ketchup with salt and pepper.
On a large rimmed baking sheet lined with foil, make 4 stacks of 2 eggplant slices. Top each with 1 cheese slice, 4 to 6 slices each of the sweet potato and beet, the remaining cheese and 4 to 6 zucchini slices. Transfer to the oven and bake for 5 minutes, until the cheese is melted and the veggie burgers are heated through.
Place the burgers on the bottom buns and top with lettuce and pickles. Spread the carrot ketchup on the top buns, close the burgers and serve.