Melt the 2 tablespoons of butter in a small saucepan, stir in the ground ginger, cinnamon, and cloves, and cook for about 1 minute until fragrant. Cool completely to room temperature.
Preheat oven to 350 degrees F. Grease and flour three cake pans and line the bottoms with parchment rounds.
Whisk the cake flour, baking powder, baking soda, and salt in a medium bowl.
In a stand mixer, beat the softened butter (3/4 cup) with the brown sugar on medium speed for 3 minutes until light and fluffy.
Add the cooled spiced butter and the molasses, mixing until fully combined.
Add eggs one at a time, mixing until just combined after each.
With the mixer on low, add the flour mixture in three additions, alternating with the buttermilk in two additions, starting and ending with flour.
Gently fold in the crushed gingersnap cookies by hand.
Divide the batter among the three pans and bake for 25 to 29 minutes, until a toothpick comes out clean.
Cool in the pans for 10 minutes, then transfer to a wire rack to cool completely before frosting.
Beat the cream cheese and butter until smooth. Add powdered sugar gradually, then mix in vanilla and salt until fluffy.
Frost between each layer, apply a thin crumb coat, refrigerate for 20 minutes, then apply the final coat of frosting and press extra gingersnaps onto the sides.
Slice and serve, or refrigerate until ready to serve.