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Preheat oven to 350 degrees and line baking sheets with Silpat or parchment paper.
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In the bowl of a stand mixer, cream together butter, brown sugar and white sugar until fluffy.
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Add egg and vanilla extract and continue mixing until combined.
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Add pumpkin and mix just until smooth.
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In a separate bowl, whisk together flour, baking soda, pumpkin pie spice, cinnamon and salt.
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Slowly add flour mixture into pumpkin mixture just until combined.
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Use a medium cookie scoop to scoop cookie dough onto baking sheets. Dough will be very soft.
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Bake cookie for about 10 minutes.
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While cookies are cooling, prepare frosting.
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In a small saucepan, melt together butter, heavy cream, brown sugar and salt.
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Whisk over medium heat until mixture comes to a boil.
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Let caramel mixture boil for about 2 minutes.
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Remove from heat and let cool for about one minute.
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In a separate bowl, whisk together caramel, sifted powdered sugar and vanilla extract and stir until smooth.
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Frost cookies by spooning caramel frosting onto cookies.
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For thicker frosting, adjust amount of powdered sugar and heavy cream as needed.