In a medium pot of salted water, bring the potatoes to a boil and cook halfway to tender, about 7 minutes. Add the broccoli and cook until the potatoes and broccoli are tender, about 5 minutes. Drain and return to the pot. Add the butter and smash; season with salt and pepper. Cover to keep warm.
Meanwhile, season the turkey with salt and pepper. In a shallow bowl, combine the panko and parmesan. In another shallow bowl, beat the egg. Dip each cutlet into the egg, then into the panko mixture, pressing to coat.
In a large nonstick skillet, heat 1 tablespoon olive oil over medium heat. Add 2 cutlets and cook, turning once, until just cooked through, 6 to 8 minutes. Repeat with the remaining 1 tablespoon olive oil and 2 cutlets. Serve the cutlets with the broccoli-smashed potatoes.