4 oz. (1/2 cup) unsalted butter, at room temperature; more for the pan
6 3/4 oz. (1 1/2 cups) unbleached all-purpose flour
2 tsp. ground ginger
1 tsp. ground cinnamon
1/2 tsp. baking soda
1/4 tsp. ground cloves
1/4 tsp. table salt
1 Tbs. minced fresh ginger
1/4 cup granulated sugar
1/4 cup packed dark brown sugar
1 large egg, at room temperature
1/2 cup unsulfured mild molasses
1/2 cup buttermilk, at room temperature

Position a rack in the center of the oven and heat the oven to 350°F. Butter the sides of an 8×8-inch square cake pan and line the bottom of the pan with parchment. In a medium bowl, mix the flour, ground ginger, cinnamon, baking soda, cloves, and salt. Set aside.

Using a hand mixer or a stand mixer fitted with the paddle attachment, beat the butter on medium speed until light and fluffy, about 1 minute. Add the fresh ginger and mix until just combined. Add both sugars and beat on medium speed until well combined and fluffy, about 1 minute. Stop the mixer and scrape down the sides of the bowl. Add the egg and mix on medium speed until well combined. Turn the mixer to low and slowly add the molasses. Add about one-third of the dry ingredients and mix until just combined. Add one-third of the buttermilk and mix until just combined. Add the remaining dry and wet ingredients in four more additions, finishing with the buttermilk and mixing until just combined after each addition. Scrape the batter into the cake pan and spread it evenly.

Bake the cake until a skewer inserted into the center comes out clean, 30 to 35 minutes. Let the cake cool completely in the pan, at least an hour.

Make ahead: The cake can be made a day ahead and stored at room temperature.