Preheat oven to 400°F Butterfly each chicken breast with a sharp knife; press open like a book.
Season both sides with poultry seasoning and pepper and pound 1/4-inch thick between 2 sheets of plastic wrap with a meat mallet. Remove plastic wrap.
Place equal amounts of prosciutto, cheese and mushrooms on each chicken breast; press down on cheese to compress.
Starting from the long side, roll up chicken tightly, then roll each in a piece of parchment paper. Twist edges of parchment to seal chicken inside.
Place seam-side down on a baking sheet; bake for 30 minutes.
While chicken is cooking, prepare sauce by melting 4 tablespoons butter in a medium saucepan. Stir in flour and cook over medium heat for about 2 minutes. Slowly whisk in chicken broth until smooth. Simmer mixture until reduced by half, stirring occasionally; remove from heat.
Grind porcini mushrooms in a blender or food processor until they form a fine powder. Set aside 1/4 cup; save remaining for another use. Whisk 1/4 cup mushrooms, sherry, sour cream and remaining 2 tablespoons butter into sauce; simmer over low heat for a few minutes more. Season to taste with salt and pepper.
Remove chicken from parchment and slice 1/2-inch thick. Arrange on 4 serving plates and drizzle with some of the sauce. Serve remaining sauce on the side.