Soak the morels in a bowl with milk or cold water for about 1 hour. Pour into a sieve, collect the liquid, rinse well and cut into small pieces.
Clean the king oyster mushrooms or other mushrooms, rub them and cut them into convenient pieces. Peel shallots and garlic and cut into fine cubes.
Cook the tagliatelle in plenty of boiling salted water according to the instructions on the package. Drain, drain and hold back some cooking water to dilute the sauce if necessary. Sweat the shallots in hot butter until translucent. Add morels and king oyster mushrooms and let them sweat for a moment. Stir in the lime juice and port wine. Add the cream and zest and season with pepper, salt and chilli. Cook for about 2 minutes, depending on the consistency you want.
Chop or pluck the basil. Mix the pasta with the sauce and basil. Keep some of the basil for decoration. Sprinkle the pasta with parmesan and basil and serve.