Season chicken generously with thyme, basil, salt, pepper, and cayenne. Heat oil in a large deep skillet or Dutch oven. Cook cubed chicken until golden and mostly cooked through, about 9 minutes.
Push chicken to the side. Add diced onion, garlic, and sliced sun-dried tomatoes. Cook 2–3 minutes until fragrant. Stir in tomato paste, cooking 1–2 minutes.
Stir in broth, heavy cream, half & half, and milk. Whisk in Parmesan until smooth. Return any chicken you set aside. Add gnocchi straight into the sauce, cover, and simmer 6–7 minutes until gnocchi are tender and chicken is fully cooked through.
Stir in spinach until wilted. Add grated Parmesan immediately before serving.
