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Cream together the butter and icing sugar very well until very smooth and creamy. No lumps of butter should be visible.
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Blend in the vanilla extract.
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Sift together the flour and corn starch.
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Blend slowly into the creamed mixture until a soft dough forms.
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Split dough into 2 equal round portions.
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Wrap the dough rounds in plastic wrap and chill for a couple of hours at least.
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You can freeze one of the cookie dough portions for later if you like.
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This dough will also last in the fridge for 3-4 days so that you can bake a few up at a time and enjoy them fresh from the oven if you like.
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To bake the cookies preheat the oven to 350 degrees F and line cookie sheets with parchment paper.
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Roll the dough out to a little less than 1/4 of an inch on a lightly floured surface. Use a 2 inch cookie cutter (or holiday shaped cookie cutters) to cut out the cookies.
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Place the cookies on the parchment paper a half inch apart. Add a half maraschino or glacé cherry to the center, or a baking gumdrop if you like.
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Bake for 15-20 minutes or until they just start to turn brown at the edges.
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Let cool for 10 minutes on the pan before transferring them to a wire rack to cool completely. Store in an airtight container.
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These cookies will freeze quite well for several weeks.