Bring 4 quarts water to a boil in a large pot.
Heat the olive oil, lemon zest, shallots, and mushrooms in a large skillet over medium heat. Add the salt and honey and cook until the shallots are softened, about 5 minutes.
Whisk in the milk and let the sauce simmer for 2 minutes.
Cook the pasta in the boiling water as per the package instructions. Reserve a 1/2 cup of the cooking liquid and drain the noodles.
Add the lemon juice to the noodles, tossing them well to combine.
Add the cheese and 1/4 cup of the reserved cooking liquid to the skillet with the shallot-lemon sauce.
Add the cooked pasta and toss it gently until well coated. Add the remaining 1/4 cup of the reserved cooking liquid, if necessary, for consistency.
Divide the dish among 4 bowls and garnish each bowl generously with pepper and fresh basil.