Rinse chicken; pat dry. Place each breast half between 2 pieces of plastic wrap. Using the flat side of a meat mallet, pound lightly into a rectangle about 1/8 inch thick. Remove wrap.
Sprinkle each chicken piece lightly with salt and pepper. Place one-fourth of the cheese 1 inch from the bottom edge of 1 chicken piece; top with one-fourth of the nuts and one-fourth of the 1 tablespoon snipped tarragon. Fold in the bottom and side of chicken; roll up jelly-roll style. Repeat with remaining cheese, chicken, nuts, and tarragon.
Unfold phyllo dough. Lay 1 sheet of phyllo dough flat on the work surface; cover remaining sheets with plastic wrap. Brush phyllo dough with some of the melted butter or margarine. Remove another sheet of dough from stack and place on top of first sheet. Brush with more melted butter. Add 2 more sheets of dough for a total of 4 sheets, brushing each sheet with butter.
Cut phyllo stack into two 14x9-inch rectangles. Discard trimmings. Place 1 chicken roll at an end of each rectangle. Roll up, jelly-roll style, folding in sides. Press ends to seal. Using a sharp knife, make 3 diagonal cuts in the top layers of roll. Repeat with remaining phyllo, butter or margarine, and chicken roll. Place wrapped chicken, seam side down, on a rack in a shallow baking pan. Brush with any remaining melted butter or margarine. Bake in a 400 degree F oven for 15 to 18 minutes or until crisp and golden.
Meanwhile, for sauce, in a small saucepan melt the 2 tablespoons butter or margarine. Add mushrooms and garlic. Cook and stir until tender. Stir in flour, the 2 teaspoons tarragon, salt, and pepper. Add the 3/4 cup milk all at once. Cook and stir until thickened and bubbly. Cook and stir for 2 minutes more. Stir in white wine or additional milk. Serve sauce with phyllo rolls. Makes 4 servings.