Preheat the oven to 350°. Butter and flour three (8-inch) round cake pans. Line each with parchment. Sift together flour, baking powder, and salt. In the bowl of a stand mixer, cream butter, sugar, and vanilla paste on medium speed until light and fluffy, 2–3 minutes. Scrape sides of the bowl, then add eggs individually, beating well after each addition. Add half of flour mixture and mix on low to combine. Scrape sides of the bowl and add half of buttermilk. Repeat process with remaining flour mixture and buttermilk. Divide batter among pans and bake until tops are golden, and a toothpick inserted in the center comes out clean, 23–25 minutes. Cool completely.