Step 1: Preheat your oven to 375°F (190°C). Line two large baking sheets with parchment paper or silicone baking mats.
Step 2: In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon, nutmeg, ginger, cloves, and salt. Set aside.
Step 3: In a large bowl, using an electric mixer, cream together the softened butter, granulated sugar, and light brown sugar until light and fluffy (about 2-3 minutes).
Step 4: Beat in the pumpkin purée and vanilla extract until just combined. Scrape down the sides of the bowl as needed.
Step 5: Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix the dough.
Step 6: Drop rounded tablespoons of dough onto the prepared baking sheets, spacing them about 2 inches apart.
Step 7: Bake for 10-12 minutes, or until the edges are set and lightly golden, but the centers still look soft. Do not overbake to ensure soft pumpkin cookies.
Step 8: Remove the cookies from the oven and let them cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely.
Step 9: While the cookies are cooling, prepare the brown butter glaze. In a small saucepan, melt the ¼ cup butter over medium heat. Continue to cook, swirling occasionally, until the butter turns golden brown and smells nutty, with small brown bits forming at the bottom. Immediately remove from heat and pour into a small heatproof bowl to cool slightly.
Step 10: Whisk the powdered sugar, milk (or cream), and vanilla extract into the slightly cooled brown butter until smooth. Add more milk, a teaspoon at a time, if a thinner glaze is desired.
Step 11: Once the pumpkin cookies are completely cool, drizzle or spread the brown butter glaze over each cookie. Let the glaze set before serving or storing. This pumpkin cookie recipe is best enjoyed fresh, but they store well.