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In a large mixing bowl, cream together butter and sugar until fluffy and pale yellow. Add egg and vanilla, mix to combine.
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In a medium bowl, combine flour, baking powder, salt, and cinnamon.
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Add dry ingredients and milk to butter mixture; alternate flour and milk additions, beginning and ending with flour mixture.
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Chill dough for 30 minutes in the fridge.
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Preheat oven to 375 degrees F.
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Place dough on a lightly floured surface, the dough will still be pretty tacky; you'll want to keep extra flour close at hand. Roll dough out to about 1/4" thick, add more flour as needed to keep dough from sticking.
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Using a floured square cookie cutter, cut out cookies and place on a Sil-Pat (or parchment) line baking sheet. Bake for 9 to 11 minutes. Let cookies set on baking sheet for 2 minutes, then remove to wire cooking rack to cool completely. Repeat with remaining dough.
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Once cookies are completely cool, make the frosting. Combine butter and powdered sugar in a large mixing bowl. Add syrup and vanilla, mix to combine. Add milk 1/2 tablespoon at a time, mixing after each addition until the frosting is the desired consistency - your want a spreadable frosting, not a drippy icing. If you add too much milk, just add a little extra powdered sugar.
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Frost cookies and sprinkle with a little cinnamon, if desired.