Position a rack in the center of your oven and preheat the oven to 350ºF. Butter or coat an 9-inch round cake pan with nonstick spray. Line the pan with a circle of parchment paper cut to size.
In a large bowl, whisk the brown sugar and eggs until pale and foamy, about 1 minute. Add the peanut butter and whisk until smooth. Add the buttermilk, oil, vanilla, and kosher salt. Whisk until smooth andemulsified.
Add the flour, baking powder, and baking soda and whisk until well-combined and smooth.
Pour the batter into the prepared pan, tap the pan gently on the counter to release any air bubbles, and smooth the top of the batter with an offset spatula.
Bake the cake until puffed and golden, and a tester inserted into the center comes out clean, 30 to 40 minutes depending on your oven. Set the pan on a rack to cool for about 15 minutes, then turn out the cake onto a wire rack to cool completely before frosting.