Cook the spaghetti in enough boiling salted water until al dente.
Slice the broccoli stems and divide into florets. Add both to the spaghetti 7 minutes before the end of cooking time.
Drain spaghetti and broccoli in a colander. Bring whipping cream to a boil.
Add half of the broccoli and puree with the whipped cream. Melt the cheese (except for 1 tablespoon) in the whipped cream while stirring and season with salt, pepper and thyme.
Toss spaghetti and remaining broccoli in sauce and serve sprinkled with remaining cheese.