Combine flour, salt and pepper in a shallow dish. Lightly dredge turkey cutlets in the flour, shaking off the excess.
Heat 1½ teaspoons oil in a large nonstick skillet over medium-high heat. Add half of the cutlets and saute until golden brown on the outside and no longer pink in the center, 2 to 3 minutes per side. Transfer to a platter and keep warm. Add the remaining 1½ teaspoons oil to the pan and cook the remaining cutlets in the same manner, transferring them to the platter when they are cooked.
Add parsnips, apples and shiitakes to the pan and cook, shaking the pan occasionally, until the vegetables begin to soften, about 2 minutes. Add vinegar and stir to scrape up any browned bits. Add broth and simmer over medium heat until the vegetables are tender and the sauce has reduced, 8 to 10 minutes. Stir in sage. Spoon the vegetables and sauce over the turkey and serve.