Preheat oven to 350°F (175°C). Grease and flour two 9-inch round baking pans or coat a 9×13-inch baking pan with butter (or pan release), tapping out any excess flour.
In a large bowl, beat 1 cup (226 g) softened unsalted butter and 2 cups (396 g) granulated sugar on medium speed until the mixture is pale and light yellow, about 2–3 minutes. Scrape down the bowl.
Add 4 large eggs to the butter-sugar mixture one at a time, mixing until each egg is fully incorporated before adding the next. Scrape the bowl.
Add the 2 large egg yolks one at a time, mixing until each yolk is fully incorporated. Scrape the bowl.
In a separate bowl, whisk together 3 cups (375 g) all-purpose flour, 1 teaspoon (2.6 g) kosher salt, and 3 teaspoons (12 g) baking powder until evenly combined.
With the mixer on low speed, add half of the dry ingredient mixture to the butter-egg mixture and mix just until combined. Add half of the 2 cups (454 g) buttermilk and mix until combined. Add the remaining dry ingredients and mix until combined. Add the remaining buttermilk and mix until just combined. Scrape the bowl as needed and avoid overmixing.
Stir in 2 1/2 teaspoons (11.75 g) vanilla extract until evenly distributed, mixing only until blended.
Divide the batter evenly between the prepared 9-inch pans or pour into the prepared 9×13 pan and smooth the top.
Bake until the cakes spring back slightly when touched and a toothpick inserted into the center comes out clean or with a few moist crumbs: about 30 minutes for two 9-inch pans or about 35 minutes for a 9×13 pan. Begin checking a few minutes before the minimum time.
Remove the pan(s) from the oven and place on a wire cooling rack. Let the cake(s) cool in the pan for 8–10 minutes. Run a knife around the pan edges, invert the cakes onto the rack, remove the pans, and let the cakes cool completely.