Melt butter in heavy large skillet over medium-high heat. Add mushrooms and saute until golden, about 6 minutes; add thyme and stir 1 minute. Season to taste with salt and pepper.
Bring chicken broth and beef broth to boil in heavy large shallow saucepan. Add sausage and simmer until cooked through, about 12 minutes. Using tongs, remove sausage. Boil broth mixture until reduced to 2.5 cups, about 15 minutes. Slice sausage into 1/2-inch thick rounds.
Add linguine to broth mixture and cook until linguine is tender but still firm to the bite (do not drain). Add mushrooms and sausage to pasta. Season with salt and pepper.
Serves 4.