Preheat the oven to 350°F (175°C).
Grease a 9-by-9 inch (23-by-23 cm) cake pan with butter and line with parchment paper, letting the excess fall over the sides.
In a small, heavy-bottomed saucepan over medium heat, melt the butter. Once melted, crank up the heat to medium-high. Continue stirring until small bronze bits begin to settle on the bottom of the pan, 3 to 5 minutes. Take the brown butter off the heat and pour into a medium heat-safe bowl to cool for 10 to 15 minutes.
Whisk the sugar and oil into the cooled brown butter until smooth. Add the eggs and whisk until thick and frothy, about 2 minutes. Add the buttermilk, sour cream, vanilla, baking powder, baking soda, and salt and whisk until well blended, 2 to 3 minutes more. (It’s okay if the mixture looks slightly curdled at this point.) Fold in the flour until just combined.
Pour half of the cake batter into the prepared pan. Smooth the top and sprinkle the brown sugar–cinnamon swirl mixture evenly over the batter. Cover with the remaining batter, smoothing the top once more.
Bake the cake in the center of the oven until the top is puffed, lightly bronzed, and bounces back when gently pressed in the center, 40 to 60 minutes. Start looking for the visual cues early, and for extra insurance against a slumping middle, use a cake tester or toothpick. The cake is done when a toothpick inserted in the center comes out with moist crumbs attached.
Let the cake cool for 10 minutes in the pan before using the parchment overhang to remove it from the pan. Place the cake on a rack to finish cooling.