Adjust oven rack to middle position and heat oven to 350ºF. Grease and flour a 9 x 13-inch baking pan; set aside. Combine cinnamon, cardamom, allspice, cloves, and nutmeg in a small bowl; reserve ½ teaspoon of spice mixture for the frosting.
In a small skillet over medium heat, melt 4 tablespoons of the butter. Cook, swirling pan constantly until butter is light brown and has a faint nutty aroma, 2-4 minutes. Add spices and continue to cook, stirring constantly, for 15 seconds. Transfer the butter mixture to a small bowl and cool to room temperature.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. In a small bowl, gently whisk eggs, yolks, and vanilla to combine. Using a stand mixer fitted with the paddle attachment, beat remaining 12 tablespoons of butter with the sugar and molasses on medium-high speed until pale and fluffy, about 3 minutes. Scrape down the sides of the bowl using a rubber spatula. Add the cooled butter mixture, ginger, and half of egg mixture; mix until incorporated, about 15 seconds. Repeat with remaining egg mixture; scrape down the sides of the bowl. Reduce speed to low and add flour mixture in three additions, alternating buttermilk in two additions, scraping down bowl as needed. Mix at medium speed until batter is thoroughly combined, about 15 seconds. Give batter a final stir by hand.
Scrape batter into prepared pan, smooth top with rubber spatula, run tip of metal spatula through batter in a zigzag motion to edges of the pan, and gently tap pan on counter to release air bubbles. Bake cake until toothpick inserted in the center comes out clean, about 32-37 minutes. Let cake cool completely in pan on wire rack, about 2 hours.