Preheat oven to 350 degrees. Pound chicken between sheets of waxed paper to thickness of 1/4-inch using meat mallet. Pat chicken dry.
Combine cheese, green onions and basil in small bowl. Season with salt and pepper. Spread cheese mixture lengthwise over half of each chicken piece. Tuck short ends in. Roll chicken up, starting at one long side, into tight cylinders. Tie ends with string to secure. Dip chicken in egg, allowing excess to drip into bowl. Roll in bread crumbs, shaking off excess. (Can be prepared 4 hours ahead. Refrigerate.) Place chicken in 8-inch square baking dish. Pour 2 tablespoons melted butter over. Bake until cooked through, about 20 minutes.
Sauce: Meanwhile, melt 1/4 cup butter in heavy large skillet over medium heat. Add mushrooms and saute until tender, about 8 minutes. Add wine and boil 3 minutes. Add stock and boil until liquid is reduced by half, about 6 minutes. Remove from heat and swirl in 4 tablespoons cold butter 1 piece at a time. Season sauce with salt and freshly ground pepper. Remove string from chicken. Cut rolls crosswise into 1/2-inch-thick rounds. Fan on plates. Serve immediately, passing sauce separately.
Recipe courtesy of Gourmet Magazine