Place oven rack in the center of your oven. Preheat oven to 350°F. Grease 2 or 3 8 or 9-inch round cake pans (see Notes) and line bottoms with parchment paper.
In the bowl of a stand mixer, add the cake flour, brown sugar, baking powder, baking soda, cinnamon, and salt. Add the butter and 1 cup of the buttermilk.
With the paddle attachment, beat mixture on low speed until moistened and combined. Increase speed to medium high and beat for several minutes until smooth and creamy, scraping down the bowl once or twice.
In a separate bowl, whisk together the remaining ½ cup buttermilk, eggs, egg yolks, and vanilla.
Add the wet ingredients to the mixer in 3 additions, mixing and scraping the bowl down after each. Mix on medium speed until smooth.
Divide batter evenly between prepared cake pans.
Bake until edges come away from the pan, and center feels firm and bounces back when gently pressed, 20-25 minutes. (Time will depend on size of pan and how many you're using. I recommend only baking 2 at a time on the center rack, if using more than 2 pans bake in 2 batches.)
Let cakes cool about 15 minutes in the pan, then turn out to a wire rack to cool completely.
I prefer to completely chill cake layers before assembling the final cake. Wrap in plastic and store in the fridge for up to 3 days, or double wrap and freeze for several months. Thaw overnight in the fridge before using.