Heat 1 tablespoon of the oil in a large skillet over medium-high. Add chicken; cook, stirring occasionally, until browned and cooked through, about 6 minutes. Transfer to a large bowl.
Heat 1 tablespoon of the oil in skillet over medium-high. Add squash, mushrooms, onion, and garlic; cook, stirring occasionally, until just tender, 10 to 12 minutes. Add water, if needed, 1 tablespoon at a time, and stir to loosen browned bits from bottom of skillet. Transfer squash mixture to bowl with chicken.
Heat remaining 1 tablespoon oil in skillet over medium-high. Add gnocchi and cook, stirring occasionally, until lightly browned, about 1 minute. Stir in broth, sage, and chicken mixture. Cover and reduce heat to medium. Cook until gnocchi is tender and most of liquid is absorbed, about 4 minutes. Stir in spinach, pesto, and vinegar; cook, stirring often, until spinach is wilted, about 1 minute. Sprinkle servings evenly with cheese.