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Preheat oven to 350ºF.
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Beat egg whites and cream of tartar at high speed with a mixer until foamy.
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Gradually add the 1/2 cup of sugar, 1 Tbs at a time, until stiff peaks form and sugar is dissolved - about 2-4 minute.
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Stir in 1/2 tsp vanilla extract and chopped pecans.
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Drop by teaspoonfuls onto baking sheets lined with aluminum foil.
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Dip a finger in the Tbs of powdered sugar and make an indentation in the center of each cookie.
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Place in the oven and immediately turn the oven OFF. Don't open the door for at least 8 hours.
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Remove from oven, and carefully peel cookies from foil.
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Combine butter, cocoa, and half and half in a small saucepan and bring to a boil over medium heat.
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Remove from heat and cool to room temperature.
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Add the 2 1/4 cups powdered sugar and beat well with an electric mixer.
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Stir in remaining 1/2 tsp vanilla.
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Just before serving, pipe cocoa mixture into the indentation of each cookie with a star tip - if you have one.