In a steamer basket, steam the squash for 10 minutes, or until tender and puree in blender or with immersion blender. Set aside
Melt the butter in a large skillet over medium heat. Saute the onion, garlic and thyme until the onion is limp and translucent. Clean, trim and slice the shiitakes. Add them to skillet. Turn the heat up to high, and saute until the mushrooms are soft and golden. Add the squash puree, chicken broth and cream and season with salt and pepper. Bring the cream to a gentle simmer, gently tossing the ingredients to mix. Grate a handful of Parmesan cheese into the sauce and continue to cook until the cream reduces slightly.
Meanwhile, (following the recipe below) cook the spaetzle in abundant boiling salted water until al dente.
Quickly drain the spaetzle and place it in the pan with the sauce. Add the walnuts and gently toss the pasta with the sauce over low heat just until it has absorbed some of the sauce. Place the pasta in a serving bowl. Dust with more Parmesan cheese and serve.
Sift flour into a large bowl. Add salt. Make a well in the center. In another bowl whisk together the eggs and water. Add to dry ingredients and beat with spatula until smooth and elastic.
Push batter through spaetzle maker into a large pot of boiling salted water and cook, uncovered, just until spaetzle is firm, floats to the top and no longer tastes of raw flour, about 3 to 5 minutes. Drain spaetzle well before using.