In a wide, shallow bowl, toss the potatoes with 1 tablespoon flour, then with the parmesan and scallions; season with salt. Place the eggs in another wide, shallow bowl and beat lightly. Place the remaining 1/2 cup flour in another shallow bowl and season with salt.
In a large, heavy nonstick skillet, heat 2 tablespoons oil over medium-high heat. Working with 1 piece at a time, coat the turkey cutlets with flour, shaking off the excess, then dip in the eggs and transfer to the potato mixture, pressing potato onto each side.
Cook the cutlets in the skillet with more oil as needed, turning once, until the potato coating is golden and crisp, about 5 minutes per side. Transfer to parchment paper or paper towels to drain.