In a small bowl, combine the milk and vinegar. Let sit for 5 minutes.
Meanwhile, in a medium bowl (if you're melting butter on the stove-top you can do this step directly in the saucepan), whisk together the melted butter and cocoa powder.
In a large bowl, add the cane sugar and eggs. Beat the ingredients, using a mixer on medium speed, for about a minute.
Stir in the milk mixture (the vinegar and milk) and chocolate mixture (butter and cocoa powder) and vanilla extract.
Add the flour by sifting it first through a sieve and into the bowl. Then, stir in the remaining dry ingredients: baking powder, baking soda, and salt. Stir just until all the ingredients are well combined.
Grease a 13x9 baking dish with extra butter and sprinkle extra flour over the pan, shaking the pan to evenly coat the surface. The goal is to keep the cake from sticking to the surface when it's time to remove it from the pan. You don't want a ton of flour, just a light dusting over the butter.
Pour the cake batter into the prepared pan.
Bake the cake for about 30-32 minutes, until a toothpick (or knife, since you'll be frosting the top no one will tell) inserted in the center comes out clean. I bake my cake in a dark pan, so if you're using a glass or light ceramic pan your baking time may need to be adjusted. As a good rule of thumb, take a peek at how the cake is doing (through the glass oven door) at around the 25 minute mark.
Remove the cake, once cooked through, and allow it to rest for about 10 minutes. Then, carefully flip the cake over on a cooling rack and allow to fully cool. You may also leave the cake in the pan if you prefer to serve it that way. Once fully cool, frost the cake with the chocolate frosting recipe, below. This cake is amazing enough to enjoy without frosting, too. This cake would also be delightful during the summer with a homemade whipped cream layer on top and some raspberries sprinkled over the whipped cream.
Store the cake, frosted or unfrosted, in the fridge for up to 4 days. The cool environment will dry out the cake (this is normal for cake), so it's best to return the cake to room temperature before enjoying a slice if you want the best texture. Or, you can store the cake in the freezer for about a month.
Make the Chocolate Frosting:
In a large bowl (wash out the bowl from making the batter and reuse it to make the frosting), add the softened butter, vanilla, and salt. Use a hand-mixer to beat the ingredients, on medium speed, until creamy and slightly fluffy.
Add the powdered sugar and cocoa powder, a small amount at a time (to ensure the dry ingredients don't go flying all over your kitchen). Beat until all the sugar and cocoa are added and the ingredients are well combined.
Finally, add the milk or heavy cream, a splash at a time, beating the frosting until you reach the desired consistency. Less milk results in a thicker frosting. More milk results in a fluffier frosting. Use as many splashes of milk (up to about 4 tablespoons) as needed.
Spread the frosting over the cooled chocolate cake.
Notes
*I tested this recipe with unsweetened cocoa powder (365 Whole Foods brand). If you'd like to use cacao powder, you're welcome to do so. This goes for the frosting and cake. **I use organic cane sugar in this recipe. If you want to use something like sucanat or coconut sugar, I think you could certainly try. I recommend sticking with the same weight measurement. I'm not sure what the overall flavor/sweetness will be like compared to using cane sugar. Dairy-Free Version: If you want to make a dairy-free cake, you could try using a neutral oil (like avocado oil) instead of the butter. Or, coconut oil--although it may impart some coconut flavor. A plant based milk (coconut milk, almond milk, or even oat milk) should work in place of the whole milk. You'll also need to find a homemade dairy-free frosting, or go with Simple Mill's Frosting (you can find this at most health food stores--it's a dairy-free store-bought option). If you're looking for a gluten-free and egg-free cake, this recipe is my recommendation. Cupcakes: You could definitely make cupcakes with this batter. I'm guessing this amount of batter would make around 24 cupcakes (?). Use muffin liners and cook the muffins for about 20-22 minutes.