Split the sausage open and break up the sausagemeat with a fork. Heat a large, heavy-based frying pan over a low heat, add the sausagemeat chunks and cook, stirring, until they are lightly golden. As the fat from the sausages heats, it will melt and stop the meat from sticking to the pan.
Increase the heat to high and stir in the chestnuts and thyme. Cook, stirring, for 1–2 minutes to colour the chestnuts, then pour in the cream and simmer gently for 1 minute until slightly thickened.
Cook the pasta in a large saucepan of salted boiling water until it is al dente: according to the packet instructions for dried pasta or for 2 minutes if using fresh pasta. Drain thoroughly, then toss into the sauce.
Place over a very low heat and add the Parmesan and milk. Season with salt and pepper. Toss gently until the sauce has thickened and the pasta is well coated with the sauce. Serve immediately with a scattering of grated Parmesan, if liked.