Whisk together flour, baking soda, cinnamon, nutmeg, cloves, and salt in a medium bowl until evenly combined. Set aside.

In a large bowl or stand mixer, beat softened butter and granulated sugar on medium-high speed for 3-5 minutes until light, fluffy, and pale yellow in color. Scrape down sides of bowl halfway through.

Add pumpkin puree and beat on medium speed until fully incorporated and smooth. Add egg and vanilla extract, beating until mixture is uniform in color and texture.

With mixer on low speed, gradually add dry ingredients in three additions, mixing only until no white flour streaks remain. Do not overmix. Dough will be soft and sticky.

Cover bowl tightly with plastic wrap and refrigerate for at least 1 hour, or up to 24 hours, to firm up dough and develop flavor.

Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper.

Using a 2-tablespoon cookie scoop, drop rounded portions of dough onto prepared sheets, spacing 2 inches apart. Place about 8-9 cookies per sheet.

Bake for 10-12 minutes until edges are lightly golden and set while centers look puffed and barely set but not browned. Cookies will appear slightly underbaked.

Let cookies rest on baking sheet for 5 minutes to finish setting, then transfer to wire rack. Cool completely, about 30 minutes, before frosting. Repeat with remaining dough batches.

For frosting: Beat softened cream cheese and butter together on medium speed until completely smooth with no lumps, about 2 minutes.

Add powdered sugar one cup at a time, beating on low speed after each addition until incorporated. Add vanilla extract and 2 tablespoons milk, then beat on medium speed. Add remaining milk 1 tablespoon at a time until frosting is smooth, spreadable, but holds its shape.

Once cookies are completely cool, use a knife or offset spatula to spread 1-2 tablespoons of frosting on each cookie. Store frosted cookies in refrigerator.