In a large saucepan, heat the olive oil over medium-high heat. Add the sliced mushrooms and cook, stirring frequently, until their moisture evaporates and they are golden brown, approximately 12 minutes.
Stir in the butter, letting it melt. Add the leeks, garlic, and 1 teaspoon of salt. Continue cooking, stirring occasionally, until the leeks soften and become fragrant, about 4 to 5 minutes.
Add the pasta, water, broth, wine, and the remaining 1 1/2 teaspoons of salt. Increase the heat to high and bring to a boil. Lower the heat to medium, allowing the mixture to simmer. Stir occasionally until the pasta is just short of al dente, based on the package's cooking time.
Stir in the cream, simmering for a few more minutes until the pasta is fully coated in the sauce. Remove from the heat. Add the grated Parmigiano-Reggiano, dill, parsley, lemon zest, juice, and black pepper. Serve the pasta in shallow bowls, garnishing with additional cheese if desired.