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Preheat oven to 350°F.
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COOKIES: Cream the butter and sugar together using an electric mixer on medium speed until creamy.
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Beat in the egg and lemon zest.
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Combine the flour, baking soda, and salt; gradually add to the creamed mixture.
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Stir in the poppy seeds with the mixer on low speed.
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Drop dough by heaping teaspoon 2" apart onto ungreased cookie sheets.
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Bake for 11-12 minutes, or until edges are lightly golden.
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Cool on sheets for 1 minute; transfer cookies to wire racks and cool completely.
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GLAZE: Combine the powdered sugar and lemon juice in a small bowl; mix well.
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Drizzle glaze over cookies in a random or decorative pattern; let stand until glaze is set, about 20 minutes.