Preheat oven to 350°. Beat egg whites and cream of tartar at high speed of an electric mixer until foamy. Gradually add 1/2 cup sugar, 1 tablespoon at a time, beating until stiff peaks form and sugar dissolves (2 to 4 minutes). Stir in 1/2 teaspoon vanilla and chopped pecans.
Drop mixture by teaspoonfuls onto cookie sheets lined with aluminum foil. Dip finger in 1 tablespoon powdered sugar, and make an indentation in center of each cookie. Place in oven, and immediately turn oven off. Do not open door for at least 8 hours. Remove from oven, and carefully peel cookies from foil.
Combine butter, cocoa, and half-and-half in a small saucepan; bring to a boil over medium heat. Remove from heat, and cool to room temperature. Add 2 1/4 cups powdered sugar, and beat well with an electric mixer. Stir in remaining 1/2 teaspoon vanilla.
Just before serving, pipe cocoa mixture (using a pastry bag fitted with a star tip) into the indentation of each cookie.