Preheat oven to 400°F. Place feta in the center of a 9-by-13-inch baking dish. Arrange mushrooms and shallots in a single layer around the feta; drizzle with ¼ cup oil and sprinkle with 1 tablespoon thyme.
Bake, uncovered, stirring once, until the mushrooms have softened, about 30 minutes.
Remove from oven; arrange 12 ounces rotini in an even layer over the mushroom mixture. Add 2 cups broth, ½ cup each wine, cream and water, 1 tablespoon garlic, ½ teaspoon pepper and ⅛ teaspoon salt; stir until combined.
Cover with foil; bake for 20 minutes. Stir the mixture and continue baking, covered, until the pasta is tender and the liquid has reduced by about half, about 15 minutes more. Stir in ¼ cup parsley and 2 teaspoons vinegar. Garnish with additional thyme, if desired.
