Preheat the oven to 350°F (177°C) and line two baking sheets with parchment paper. Rotate sheets halfway through baking for even color.

Whisk together flour, baking soda, ground ginger, cinnamon, cloves, and salt in a medium bowl until homogeneous.

In a stand mixer with paddle attachment, cream butter, granulated sugar, and brown sugar on medium-high until pale and fluffy, about 3 minutes.

Beat in the egg and molasses on low speed until just combined, scraping the bowl as needed.

Mix in the dry ingredients on low speed until a soft dough forms, about 20–30 seconds; avoid overmixing.

Scoop 1.5 tablespoons of dough onto the prepared sheets, spacing each by 2 inches.

Bake for 9–11 minutes, until edges are set and centers look slightly underdone. Let rest on the sheet for 5 minutes, then transfer to a wire rack to cool completely.

Beat softened butter on medium-high until creamy, about 2–3 minutes.

Add powdered sugar, cinnamon, nutmeg, eggnog, vanilla, and salt. Mix on low until sugar is absorbed.

Gradually add additional powdered sugar up to 4 cups (480g) until the frosting reaches spreadable consistency.

Increase speed to medium-high and beat until light and fluffy, about 2 minutes.

Flip half the cookies so the flat sides face up. Pipe or spread about 2 tablespoons of frosting onto each, then sandwich with remaining cookies.