First, remove the moisture from the pumpkin puree. Spread the pumpkin puree in a thin layer on several layers of paper towels. Use additional paper towels to blot the pumpkin puree until most of the moisture is removed and the pumpkin puree is dry. The pumpkin puree will reduce by half, so you should have ⅓ cup of pumpkin puree remaining. Set aside.
Next, make the brown butter. Place the unsalted butter in a medium saucepan on the stove over medium-low heat. Heat until the butter is fully melted. Once the butter is melted, stir the butter constantly and continue to cook until it foams up and the milk solids turn a golden brown color. Pour the brown butter into a heat-proof bowl and cool to room temperature before making the cookie dough.
Preheat the oven to 350℉ (177℃). Line baking sheet pans with silicone baking mats or parchment paper.
In a large mixing bowl, whisk the light brown sugar, granulated sugar, and cooled melted brown butter. Add the egg, egg yolk, and vanilla extract and whisk until smooth. Add in the dried pumpkin puree and mix well until it is fully combined.
In a separate mixing bowl, whisk together the all-purpose flour, pumpkin pie spice, baking soda, and salt. Add the dry ingredients into the wet ingredients and mix well with an electric hand mixer or spatula until the ingredients are combined and the cookie dough forms. Fold in the semisweet chocolate chips.
Portion the cookie dough using a large cookie scoop (3 tablespoons) and place the cookie dough balls on the prepared baking sheet, allowing room for spreading. Press additional semisweet chocolate chips into the top of the cookie dough, if desired.
Bake the cookies for 10 to 12 minutes, until the edges of the cookies are set and the centers are slightly underdone. Cool the cookies on the pan for 2 minutes and then transfer the cookies to a wire rack to cool completely.