1. Preheat oven to 500°.
2. Remove giblets and neck from turkey; discard. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Combine 1 tablespoon oil, thyme, 1/2 teaspoon salt, 1/2 teaspoon pepper, and cheese in a small bowl. Rub cheese mixture under loosened skin over flesh. Place 1 pear and 1 onion in cavity. Lift wing tips up and over back; tuck under turkey. Tie legs together with kitchen twine. Place turkey on the rack of a roasting pan coated with cooking spray. Arrange remaining 3 pears and remaining onion, cut sides down, in a roasting pan coated with cooking spray; place rack with turkey in pan. Bake at 500° for 30 minutes. Pour 1 1/2 cups stock over turkey; reduce heat to 350°. Bake at 350° for 1 hour. Pour 1 1/2 cups stock over turkey; bake for 10 minutes. Brush 1 1/2 tablespoons walnut oil over turkey; bake 20 minutes. Brush remaining 1 1/2 tablespoons walnut oil over turkey; bake 10 minutes or until a thermometer inserted in the thickest part of the thigh registers 160°. Remove from oven. Place turkey, breast side down, on a rimmed baking sheet coated with cooking spray. Let stand 30 minutes.
3. Strain pan drippings through a sieve into a bowl; discard solids. Place a zip-top plastic bag inside a 2-cup glass measure. Pour pan drippings into bag. Let stand 10 minutes (fat will rise to the top).
4. Seal bag; carefully snip off 1 bottom corner of bag. Drain drippings into a saucepan, stopping before fat layer reaches opening; discard fat. Combine remaining 1 1/2 cups stock and flour in a bowl, stirring with a whisk. Add stock mixture and walnuts to pan; bring to a boil. Reduce heat, and simmer 10 minutes or until mixture thickens. Remove from heat; stir in remaining 1/2 teaspoon salt and remaining 1/2 teaspoon pepper. Pour mixture into a blender. Remove center piece of blender lid (to allow steam to escape); secure lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Process until smooth. Serve gravy with turkey.