Heat water to a boil for the pasta, salt water, drop in pasta and cook to al dente.
Wipe mushrooms clean with damp towel. Remove woody stems. Thinly slice the mushrooms.
Heat a large deep skillet with extra-virgin olive oil over medium-high heat. Add the mushrooms and cook until deeply golden and tender, 10 minutes.
While mushrooms cook, halve the leeks lengthwise. Cut off a couple of inches from tough tops. Trim off root end. Thinly slice the leek then vigorously wash in a large bowl of cold water, separating all the layers to free of grit. Let leeks sit a few minutes then lift out of the water and dry on kitchen towel.
Add leeks to mushrooms along with garlic and season with salt, pepper and fresh thyme. Cook 3 to 4 minutes more then add wine, reduce half a minute then stir in cream and heat through. Toss pasta with mushrooms and adjust seasoning. Top with grated cheese at the table.