Brown the butter by melting it in a light-colored skillet. Cook the butter, stirring constantly, until it becomes foamy and the milk solids separate and settle on the bottom of the pan. Continue to cook until some foam subsides and the milk solids are brown and the butter smells intensely nutty. Immediately transfer the butter to a glass bowl and place it in the refrigerator or freezer until it is no longer warm to the touch, but still liquid.
Cook the pumpkin puree in a small saucepan over medium heat for 10 to 15 minutes, stirring regularly, to remove excess moisture. The puree should be thick and paste-like. Transfer the puree to a bowl and place it in the fridge to cool to room temperature.
Preheat your oven to 350°F. Line baking sheets with silicone baking mats or parchment paper and set them aside.
Place the cooled brown butter, brown sugar, and granulated sugar in a large mixing bowl and beat with an electric handheld mixer for 3 to 4 minutes until light and creamy.
Add the cooled pumpkin, egg, egg yolk, and vanilla, and beat until combined.
Add the flour, cinnamon, baking soda, salt, allspice, nutmeg, cloves, and cardamom and beat until no more streaks of dry flour are visible. Fold in the chopped chocolate with a silicone spatula.
Use a 1.5 tablespoon cookie scoop to portion out cookie dough balls on the lined cookie sheets about 2-inches apart. Press the remaining chocolate into the tops of each cookie and bake the cookies in batches for 10 to 12 minutes until the edges are golden brown and the tops are no longer raw. The centers should appear slightly underbaked. Sprinkle flaky salt on top of the warm cookies.
Allow the cookies to cool on the baking sheet for 5 minutes then transfer to a cooling rack to cool completely.