Position a rack in the center of the oven and preheat the oven to 350°F. Generously spray three 8-inch cake pans with cooking spray and line the bottom(s) with parchment paper (cut to fit). Spray the parchment, too.
In a medium bowl, whisk together the flour, brown sugar, baking powder, baking soda, and salt.
In the bowl of a stand mixer fitted with the whisk attachment, combine the peanut butter, buttermilk, water, oil, eggs, and almond extract. Whisk on low until the mixture is smooth and caramel in color, about 1 minute. Gradually add the dry ingredients, using a rubber spatula to scrape down the bottom and sides of the bowl as necessary. Take extra care here—a small amount of peanut butter always sticks to the bottom. Whisk on low until just combined. Pour the batter into the prepared pans. Note that this recipe makes around 54 ounces of batter; pour 18 ounces into each cake pan and bake for 35 to 40 minutes. When done, the top of the cakes should bounce back when gently pressed and a skewer inserted into the center of each cake should come out with a few crumbs attached. Cool completely in the pans on a wire rack.
Once cool, turn each cake out of its pan over a sheet of plastic wrap. Peel off the parchment paper and discard. Wrap each cake in the plastic wrap, ensuring that very inch of its surface area is covered tightly. Freeze overnight.