Heat oil in a large saucepan over medium-high. Add mushrooms and cook, stirring often, until all liquid has evaporated, about 12 minutes.
Add butter, stirring until melted. Add leeks, garlic, and 1 teaspoon of the salt and cook, stirring often, until leeks are tender, 4 to 5 minutes.
Stir in pasta, water, stock, wine, and remaining 1 1/2 teaspoons salt. Increase heat to high and bring to a boil. Reduce heat to medium and simmer, stirring occasionally, until about 3 minutes short of al dente (check cook time on package).
Stir in cream and simmer until sauce evenly coats pasta, about 3 minutes. Remove from heat. Stir in cheese, herbs, lemon zest and juice, and pepper. Divide pasta evenly among shallow bowls and serve topped with cheese.