Place a ziplock bag inside a 2-cup glass measure. Pour pan drippings into bag; let stand 10 minutes. Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into a small bowl, stopping before fat layer reaches opening. (You should have about 1 1/2 cups drippings.) Discard fat.

Place roasting pan from turkey or a large saucepan on a burner over medium-high. Add oil; swirl. Add neck. Add heart and gizzard to pan, if using; cook 3 minutes or until browned, stirring frequently. Combine 1/2 cup stock and flour in a bowl. Add turkey drippings, flour mixture, remaining 2 cups stock, pepper, and salt to pan, scraping to loosen browned bits. Strain stock mixture into a medium saucepan; bring to a boil. Reduce heat, and simmer 8 minutes or until reduced to 3 1/2 cups. Stir in thyme.