Melt 2 tablespoons of the butter in a medium skillet over medium heat and saute the onions, adding a pinch of salt, until golden brown, about 15-20 minutes. Transfer to a bowl and set aside.Heat the remaining tablespoon of butter in the skillet and add the mushrooms. Saute for about 5-6 minutes and then add the wine, parsley, and a pinch of salt, deglazing the bottom of the pan. Simmer until the wine is evaporated. Transfer the mushrooms to a bowl, reserving two tablespoons for the cream sauce.Add a tablespoon of oil to the skillet and once it's hot add the chicken breasts. Fry briefly on both sides until browned. Transfer the chicken to a plate and set aside.
Place the puff pastry sheets on a lightly floured work surface. On each pastry sheet, spoon half of the blue cheese mixture in the center of each sheet followed by half of the onions and half of the mushroom mixture. Place the chicken breasts on top.Brush some of the egg mixture along each side of the puff pastry sheets. Carefully fold the corners of the puff pastry sheets up over the chicken and press to seal. Brush with the egg mixture. Place the each chicken Wellington seam-side down onto a baking sheet. Brush all sides with the egg mixture.Bake for 30 minutes or until the pastries are a rich golden brown.
For the Parmesan Cream Sauce:While the chicken Wellingtons are baking, melt the butter in a small saucepan over medium-high heat. Whisk in the flour until blended and smooth and cook for a minute, whisking constantly. Add the heavy cream, whisking continually until smooth and thickened. Add the reserved mushroom mixture and simmer for two minutes. Reduce the heat to low and whisk in the Parmesan cheese until dissolved.
Cut each Chicken Wellington in half, place a half on each plate, and serve with the sauce. See side dish recommendations in blog post.