First, preheat the oven to 350°F (175°C). Grease and line an 8×8-inch cake pan to prevent sticking and make removal easy. This initial step sets the stage for a smooth baking experience.

Next, brown the butter over medium-low heat until it gives off a nutty aroma and browned specks form. Remove from heat and stir as it cools to avoid any burning, which brings out that signature flavor. In a large bowl, whisk together the flour, baking powder, and salt to combine the dry ingredients evenly.

Using a mixer, beat the browned butter with sugar, neutral oil, and vanilla bean seeds for about 1 minute until well mixed. Add the eggs and beat until fully combined, creating a creamy base. Then, alternately add the dry ingredients and milk to the wet mixture, mixing well and scraping the bowl to remove any lumps for a smooth batter.

Pour the batter into the prepared pan and bake for 23 to 25 minutes, or until a skewer inserted in the center comes out clean. Cool the cake in the pan for 5 to 7 minutes, then transfer it to a rack to cool completely, about 1 hour.

Once the cake is cool, prepare the frosting. Beat the butter until smooth, then sift in the powdered sugar gradually. Add the maple syrup, heavy cream, and a pinch of salt, and beat for 2 minutes to get a light and fluffy texture. Spread it evenly over the cake, decorate as you like, and slice to serve.