Melt 1 tablespoon butter in a large ovenproof skillet over medium heat. Add the squash and cook, stirring, until slightly soft and golden, about 8 minutes. Add the garlic, sage, red pepper flakes and 1 teaspoon salt; cook until the garlic is soft, about 2 more minutes.
Preheat the broiler. Add the chicken broth to the skillet. When it starts to simmer, stir in the kale and cook until it wilts slightly, about 2 minutes. Add the gnocchi, stirring to coat. Cover and cook until the gnocchi are just tender, about 5 minutes.
Uncover and stir in 1/4 cup parmesan and the remaining 1 tablespoon butter. Sprinkle with the remaining 1/2 cup parmesan; transfer to the broiler and cook until golden and bubbly, about 3 minutes.