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Grind 1/2 cup of the coconut in food processor.
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Chop remaining coconut and set aside.
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Add flour, 1/4 cup of the confectioners' sugar and the salt. Pulse to blend. Add butter and vanilla; pulse until dough comes together.
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Heat oven to 350°F Roll dough into 1-inch balls. Place dough balls 1 inch apart on an ungreased baking sheet.
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Bake cookies until firm but tender, 15 minutes.
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Remove to rack; let cool completely.
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In small bowl, stir together remaining 1 cup confectioners' sugar and enough milk until smooth but still thick.
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Dip cookies in glaze (about 1/2 teaspoon for each), letting it drip down sides.
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Dip in chopped coconut and set aside for glaze to dry.