Preheat oven to 350 degrees then coat an 8" x 12" cake pan with butter and flour or a good-quality baking spray. In a large bowl or the bowl of a stand mixer fitted with the whip attachment, beat sugar, butter and oil together until well combined.
Add eggs, salt and vanilla to bowl then beat on high speed for about 5–6 minutes until very creamy, like the consistency of mayonnaise. Remove from mixer if using. The rest of the mixing will be done by hand.
Sift cake flour and baking powder together in a separate bowl or on a sheet of parchment paper. Add ⅓ of the dry ingredients to the bowl and use a rubber spatula to gently fold the flour into the wet ingredients. When flour is mostly incorporated, add half of the milk and gently stir to combine.
Repeat process with an additional ⅓ of the dry ingredients and the remaining milk—ending with folding in the final ⅓ of the dry ingredients. Fold carefully to retain as much fluffiness in batter as possible as this will make for a lighter cake.
Transfer batter to prepared cake pan and bake 30–40 minutes or until barely firm in the center and a toothpick tests clean. Remove from oven and cool completely on a rack.
When cake is cool, make glaze by melting butter in a small skillet over medium heat and place a small bowl nearby to transfer the butter into. Continue cooking butter over medium heat, watching carefully, until it becomes foamy.
When foam begins to subside, swirl pan gently and watch closely as butter begins to brown. At first sight of browning, remove from heat, and continue cooking with heat retained in the pan. Continue to swirl pan until butter reaches a very light amber color and smells nutty. Transfer immediately to bowl.
Add powdered sugar, vanilla and milk and stir to combine. Adjust consistency with additional milk or powdered sugar to form a pourable glaze. Pour glaze over cooled cake and spread evenly to edges. Allow glaze to set slightly before cutting and serving.
One Last Thing
A good-quality vanilla extract will do just fine for this recipe, but vanilla bean paste has much more flavor and is cheaper than using fresh vanilla beans!