Preheat oven to 325°. Grease an 8" × 8" × 2" cake pan with 1 tsp. of the butter, then lightly flour pan with 1 tbsp. of the flour, discarding any excess. Set aside.
Sift remaining flour, cinnamon, nutmeg, baking powder, cloves, and salt together into a medium bowl. Repeat and set aside.
Beat 1½ cups of the butter in a large bowl, using a wooden spoon, until it is soft and very pliant, about 4 minutes. Add sugar and beat together until pale and fluffy, 5–6 minutes. Beat in whole eggs, one at a time, until mixture is light and fluffy, about 4 minutes. Add flour mixture in 3 equal parts, stirring with a wooden spoon after each addition, to make a smooth batter. Add egg yolks and vanilla and continue to mix until well combined, about 30 seconds more.
Pour one-quarter of the batter into prepared pan, spreading it evenly over bottom, and bake until golden brown and a toothpick inserted in the middle of the cake comes out clean, about 20 minutes. Remove cake from oven, brush cake with one-third of remaining butter, and carefully top with another quarter of the batter; repeat the baking process. Repeat the process 2 more times to use up all the batter and remaining butter. The cake will have 4 layers.
When its final layer is golden brown and a toothpick inserted into it comes out clean, remove cake from oven and set aside to let cool for 10 minutes. Run a knife carefully around edges of pan to loosen cake, then invert it onto a large plate. Invert again and let cool completely.