To make the cake, preheat your oven to 350F. Butter or spray with oil three 8-inch or 9-inch round cake pans. Line the pans with parchment paper rounds.
Add the flour, baking soda, baking powder, and salt to a large mixing bowl. Mix to combine.
In a separate mixing bowl, add the non-dairy milk and vinegar. Allow them to sit for a few minutes. Then, add the sugar, vegetable oil, and vanilla extract. Whisk the wet ingredients together until they're mixed.
Pour the wet ingredients into your dry ingredients. Use a spatula to fold everything together. Avoid any streaks of flour, but little clumps in the batter are OK. Pour the batter as evenly as you can into the three cake pans. Bake the cakes for 25-30 minutes, or until golden brown at the edges and set in the center.
Transfer the cakes to a cooling rack. After 30 minutes, remove the cakes from their pans (I like to use pans with a removable bottom—it makes this step easy). Cool the cakes on the cooling rack until they are completely cool, at least a couple hours.
To make the frosting, place the butter in a stand mixer fitted with the paddle attachment. Mix on medium speed for 2 minutes, or until the butter is fluffy. Stop the mixer, then add the sugar. Cover the mixer with a tea towel. Mix the frosting on low speed for 2 minutes. Remove the tea towel and add the salt, vanilla, and non-dairy milk. Beat on medium speed for another 3-4 minutes, or until the frosting is very fluffy again. Beat in the white vinegar, then stop the mixer.
Trim the tops and sides of the cakes to make them even for frosting if you wish. (If you do this, save the scraps for crumbling onto fruit, vegan yogurt, or ice cream, or just for snacking.) Place one layer of cake onto a cake stand and cover the top with frosting (about 1 cup). Place another layer on top and frost. Add the final layer of cake and frost it. Proceed to frost sides, maintaining a thin layer of frosting to start with—this is your crumb coat. Refrigerate the cake after the crumb coat application for 20-30 minutes.
Use the rest of the frosting to cover the entire cake, making some nice swirls as you go. Make any additional decorations you like. Slice and enjoy your cake!
In place of either cake flour or an all-purpose flour + cornstarch mix, you can use all unbleached, all-purpose flour: 3 1/2 cups (420 g).