Heat a large pot of salted water and cook pasta according to box. Reserve about 1/2 a cup of pasta water. You probably won’t need all of it, but good to have as needed as you build your sauce.
Add olive oil to a large skillet, and add sliced shallots. Cook for about 3-4 min or until they start to soften. Add in sliced shiitake mushrooms, followed by butter and thyme. Cook for another 5-7 min or until mushrooms soften and fully coated in butter.
Add balsamic and reduce heat to low. Cook stirring often for about 10-12 min or until the balsamic is reduced by more than half and mushrooms are a deep dark red color. Add in greek yogurt, heavy cream and pasta water. Cook on low until everything is creamy and combined. Season to taste and add salt and pepper as needed. Add pasta to the pan, along with parmesan cheese. Still until everything is combined, creamy and smooth.
Serve with more parmesan and parsley for garnish. Enjoy