Position the oven rack in the lower 1/3 of the oven. Preheat the oven to 425° F.Make the dressing. In a bowl or glass jar, whisk together the olive oil, balsamic, fig preserves, garlic, thyme, salt, pepper, and chili flakes. On a rimmed baking sheet, toss together the halved brussels sprouts and shallots. Pour 2/3 of the dressing over the sprouts. Lay the brussels sprouts cut side down. Bake on the lower oven rack until deeply browned, 20 to 25 minutes. Remove the brussels sprouts from the oven and toss with the cheese. Arrange the roasted brussels sprouts on a plate and top with pomegranates. Spoon over the remaining dressing, if desired.