Cook the tagliatelle in a large pan of boiling salted water -- the bigger the pan, the less chance of sticking -- giving it a stir occasionally. This can take between eight to 13 minutes from boiling -- check the packet instructions.
Cut the smoked salmon into long strips and mix with the basil. Pour the cream into a pan and bring to the boil, then boil for one minute until it begins to thicken. Stir in four tablespoons of the Parmesan and season with pepper.
Drain the pasta and toss with the Parmesan cream, then fold in the smoked salmon and basil mixture until nicely combined. Divide among warmed, wide-rimmed bowls and garnish with extra basil to serve.