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In the bowl of a stand mixer, cream the butter, brown sugar and granulated sugar on medium low speed until light and fluffy, about 3-5 minutes.
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Add in the sourdough starter, vanilla extract and maple extract and mix until combined.
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Add in the flour, salt, and walnuts and mix.
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Place a large sheet of plastic wrap on the counter and place the dough in a line down the center.
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Use the plastic wrap to roll/squeeze the dough into a log about 1.5″ wide. (Rolling it like you are making a play dough worm helps!)
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Wrap the dough well in the plastic wrap and place in the fridge to chill for at least 1 hour.
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Preheat oven to 350 F
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Prepare two baking sheets by lining with parchment paper or a silpat mat.
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Unwrap the chilled dough and use a knife to cut 1/4″ thick slices of dough.
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Place the slices 2″ apart on the baking sheets.
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Bake in the preheated oven for 10-12 minutes, until the edges are just starting to turn golden and the tops are firm.
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Allow to cool 10 minutes on the pan before moving to a wire rack to finish cooling.
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When the cookies are cool, mix together the ingredients for the icing until smooth and drizzle over the cookies.
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Allow the cookies to sit on the wire rack for at least 30 minutes to allow the icing to set.