Remove thyme leaves from stems. Grate cheese (⅓ cup).
Season chicken with ½ teaspoon salt and ¼ teaspoon pepper; coat chicken with flour on both sides, shaking off any excess (wash hands).
Preheat large sauté pan on medium-high 2–3 minutes. Place oil and butter in pan; cook chicken 2–3 minutes on each side until browned and 165°F. Remove chicken from pan and cover to keep warm.
Place mushrooms, garlic, thyme, and remaining ½ teaspoon salt and ¼ teaspoon pepper in pan; cook and stir 2 minutes. Cut pasta in half.
Stir in stock, half-and-half, and pasta; cook and stir 3–4 minutes until pasta is tender. Stir in cheese and spinach; cook 1 minute or until cheese melts. Slice chicken and serve over pasta.