Melt butter in a large skillet over medium-high heat. Cook mushrooms, onion, celery and thyme until onion is very soft and golden brown, about 5 minutes. Stir in flour and cook, stirring constantly, about 4 to 5 minutes or just until light golden brown.
Gradually whisk in 2 cups broth and wine. Cook, stirring frequently with a whisk, about 3 minutes or until thickened and smooth. Let cool. Refrigerate gravy base and remaining 2 cups broth up to 2 days before serving.
Transfer roasted turkey to a platter to rest. Skim fat from pan drippings. Add pan drippings to refrigerated gravy base in a medium saucepan; heat over medium heat, stirring frequently with a whisk. Add reserved broth 1/4 cup at a time until desired consistency is reached. Simmer until heated through and season to taste with pepper.