Place the butter in a light-colored saucepan over medium heat. Cook, stirring often, until it foams and develops golden-brown specks with a toasted aroma. Immediately pour the butter and browned bits into a large heat-safe bowl. Cool for 10 to 15 minutes.
Spread the pumpkin purée over several folded paper towels. Press more paper towels over the top to absorb the liquid. Continue blotting until the pumpkin is thick and has reduced to approximately 1/4 cup.
Add the brown sugar and granulated sugar to the cooled brown butter. Whisk until combined.
Add the blotted pumpkin, egg yolk, and vanilla. Whisk until smooth.
In a separate bowl, whisk together the flour, baking soda, pumpkin pie spice, cinnamon, and salt.
Add the dry ingredients to the wet ingredients. Fold gently with a rubber spatula until almost combined.
Fold in the chocolate chips. Stop mixing as soon as no dry flour remains.
Cover the dough and refrigerate for 30 to 60 minutes.
Preheat the oven to 350°F. Line two baking sheets with parchment paper.
Scoop the dough into 2-tablespoon portions. Roll into balls and arrange them about 3 inches apart.
Press a few extra chocolate chips into the tops.
Bake one sheet at a time for 10 to 13 minutes. The edges should look set and the centers should still look slightly soft.
Let the cookies cool on the pan for 10 minutes before transferring them to a wire rack.
Add flaky sea salt while the cookies are still warm, if desired.