Preheat the oven to 350°F (180°C). Spray a 6" round cake pan with nonstick cooking spray and line the bottom of the pan with a round of parchment paper. Spray the parchment paper with cooking spray and then set aside.
Make the cake: In a medium bowl using an electric hand mixer, beat together the butter and sugar until well combined, about 1-2 minutes.
Add the egg, vanilla bean paste, and vanilla extract, and beat until combined, about 30 seconds.
In a small bowl, whisk together the cake flour, salt, and baking soda. Add half of the dry ingredients to the batter and beat for just a few seconds before stirring in 3 tablespoons of the buttermilk. Add the remaining dry ingredients and beat, then beat in the remaining buttermilk.
Pour the batter into the prepared pan, smooth out the top, and bake on a small sheet pan for 35-37 minutes. To test for doneness, insert a toothpick into the center of the cake; if it comes out clean, it’s done. Allow cake to cool completely in the pan set on a wire rack. The cake must be completely cool before frosting.
Make the frosting: In a medium bowl using an electric hand mixer, beat the butter on medium speed until creamy, about 2 minutes. Add the powdered sugar, vanilla bean paste, and milk. Beat on low speed for 30 seconds, then increase to medium-high speed and beat for 2 full minutes. Add up to 1/4 cup more powdered sugar if frosting is too thin or another tablespoon of milk if frosting is too thick. Add a pinch of salt if frosting is too sweet. (I add 1/8 teaspoon salt.)
Decorate: Using a large serrated knife, slice a thin layer off the top of the cake to create a flat surface. Discard (or snack on while decorating!). Place the cake layer upside down on your cake stand, cake turntable, or serving plate. Using an offset spatula, if you have one (a butter knife if you don’t), frost the cake on the top and sides with a very thin layer of frosting. Don't worry if there are some visible crumbs in this first layer of frosting; that's what the crumb-coat frosting layer is intended to do: lock those crumbs down. Place the cake layers in the freezer for 15 minutes, or in the refrigerator for 30 minutes. This will allow the thin layer of frosting to harden, sealing in the crumbs. Once chilled, spread the remaining frosting all over the top and sides in an even layer (or pipe it on in decorations if you're feeling fancy.) Decorate with sprinkles if desired.
Refrigerate for at least 30-45 minutes before slicing. This helps the cake hold its shape when cutting. Cover leftover cake tightly and store in the refrigerator for up to 5 days.
If you don't have buttermilk on hand, use 6 tablespoons of milk (whole or nut milk) and 1/2 teaspoon of vinegar (apple cider or white).