Bring a large pot of salted water to a rolling boil for the pasta.
In a separate small saucepan, combine the cream, cooked chestnuts, and thyme sprig over medium heat.
Bring the mixture just to a boil, then immediately remove from the heat and let it steep for 10 minutes to infuse the flavors.
Remove the thyme sprig. Carefully transfer the chestnut and cream mixture to a blender or food processor.
Add a pinch of salt and pepper, then purée until completely smooth and creamy.
While the sauce is steeping, cook the pasta in the boiling water according to package directions until al dente.
Before draining, reserve about 1 cup (240 ml) of the starchy pasta water.
Drain the pasta and return it to the pot.
Pour the chestnut cream sauce over the hot pasta and toss to coat everything evenly. If the sauce is too thick, add a splash of the reserved pasta water until it reaches your desired consistency.
Taste and adjust seasoning with more salt and pepper if needed.
Serve immediately, garnished with a generous sprinkle of freshly grated Parmesan cheese.
Don't skip reserving the pasta water! The starch in the water helps the sauce cling to the pasta and creates a silkier texture.