In a large bowl, whisk together butter, brown sugar, egg yolk, vanilla, pumpkin, and instant coffee granules.
In a medium bowl, whisk together salt, cinnamon, pumpkin pie spice, ginger, clove, baking soda, and flour.
Add the dry ingredients to the wet ingredients, and stir until just combined.
Chill cookie dough in the refrigerator for at least 2-3 hours, until firm, and preferably overnight. Do not skip this.
Line cookie sheets with parchment paper or silicone baking mats.
Scoop the dough by tablespoonfuls, and shape into balls.
To make the cookies fatter and taller, rather than thin and flat, shape the balls into more of a tall cylindrical shape.
Bake at 350 degrees for 10-12 minutes, or until edges are golden, and the centers appear set. If you tap the top of the cookie, it should only slightly depress. You want to underbake these slightly so that they remain chewy in the center.
Remove from heat, and allow to cool on cookie sheets for 2-3 minutes before removing to wire racks to cool completely. They will continue to cook a bit on the sheets.
For the Brown Butter Drizzle
Place the powdered sugar in a medium bowl, and set aside.
Melt butter in a small saucepan over low-medium heat, stirring frequently, until it foams, the foam dissipates, it turns a deep golden brown, and develops brown flecks and a nutty fragrance, about 10 minutes or so. Toward the later stages, around the time it begins to foam, watch the brown butter very closely. It can go from \'brown\' to \'burned\' very quickly.
Remove from heat, cool 2-3 minutes, and carefully pour over powdered sugar.
Add milk and vanilla, and vigorously whisk the mixture until smooth. If the icing seems to seize or separate, you just need to really mix it to emulsify it. You can use an electric mixer if preferred. But don\'t give up thinking it\'s ruined, it just needs to be emulsified, so give it a serious mixing! Add additional milk or powdered sugar, as needed, to achieve desired consistency.
Drizzle brown butter icing over each cookie. Allow to set, and enjoy!