ends trimmed, peeled, and cut into small dice
ends trimmed and separated into leaves
Heat the oven to 450°F and arrange a rack in the middle. Heat 1 tablespoon of the oil in a medium saucepan over medium heat. When it shimmers, add shallots and cook, stirring occasionally, until golden brown, about 10 minutes. Remove from heat and carefully stir in wine, vinegar, water, sugar, and bay leaf. Simmer, stirring occasionally, until shallots are very tender and liquid is thick and syrupy, about 30 minutes. Discard bay leaf, taste sauce, and add more salt or freshly ground black pepper, as desired. (Sauce can be made up to 2 days ahead. Store in an airtight container.)
Meanwhile, combine mushrooms with remaining 2 tablespoons oil and season well with salt and freshly ground black pepper and toss to coat. Spread mushrooms in a single layer on a rimmed baking sheet and roast, stirring occasionally, until mushrooms are shrunken, and deeply browned, about 30 to 40 minutes.
Remove mushrooms from oven, add to agrodolce sauce, and stir to coat. Let cool slightly, at least 15 minutes. (The mushroom sauce can be made up to two days in advance; store covered in the refrigerator. When ready to use, let sit at room temperature 30 minutes and stir before serving.) To assemble, place 1 tablespoon of the mushroom mixture in each endive leaf and serve.