In a heavy large skillet over medium-high heat, warm 1 tablespoon of the oil. Sprinkle the chicken with salt and pepper. Add the chicken to the skillet and brown, about 1½ minutes per side. Using tongs, transfer the chicken to a plate. Reduce the heat to medium; add the onion and sage, and sauté until tender, about 5 minutes. Add the squash and turnips, and stir to coat with oil. Add 1½ cups of the broth. Bring to a boil, reduce the heat to medium-low, cover, and simmer until the chicken is tender, about 25 minutes.
Meanwhile, in a large nonstick skillet over medium-high heat, warm the remaining 1 tablespoon oil. Add the mushrooms, sprinkle with salt and pepper, and sauté until tender, about 3 minutes. Remove from the heat. Chop the greens from 1 bunch of the turnips (reserve the remainder for another use).
Add the mushrooms and chopped turnip greens to the chicken. Cover and simmer until the greens wilt, about 5 minutes. In a small bowl, place the cornstarch; gradually add the remaining 1 tablespoon chicken broth, mixing until smooth. Add to the chicken mixture, mix it in, cover, and simmer until the sauce thickens, about 2 minutes. Taste and adjust the seasoning, and serve right away.