Begin by melting the butter in a saucepan over medium heat. Continue to cook it until the milk solids turn a deep amber color and release a nutty aroma. Once done, pour the butter into a bowl and chill it until it reaches a soft, solid state, which should take about 45-60 minutes.
While the butter is chilling, toast the cake crumbles by placing them in an oven preheated to 325°F (163°C) for 5-8 minutes, or until they are golden-brown and crisp.
Once the brown butter has solidified, beat it on medium-high speed until it becomes pale and voluminous, which should take about 3-5 minutes.
Gradually add the sifted confectioners' sugar, vanilla bean paste, and fine sea salt while mixing on low speed. After the sugar is incorporated, increase the speed to medium.
Finally, add the heavy cream one tablespoon at a time until the frosting achieves a silky texture with sharp peaks.
