In a saucepan, melt the butter over medium heat. Continue to cook the butter, swirling the pan occasionally. It will hiss and spatter a bit, as the water evaporates, and the milk solids settle onto the bottom of the pan. When it's nearly ready, a layer of foam will form on top of the butter, and if you nudge the foam aside, you'll see nutty, golden brown milk solids on the bottom of the pan (if they turn dark brown or black, you've gone too far and burned the butter). The browning process might take about 5-10 minutes.
Remove the pan from the heat and use a spatula to scrape all the browned milk solids off the bottom of the pan. Pour the browned butter and all the browned bits into a bowl.
Let cool until the butter returns to a spreadable solid consistency. To speed this up, put the bowl in the refrigerator for 1-2 hours and stir the butter every 20 minutes or so until it's firmed up but still spreadable. (I usually brown the butter in the evening, then just let it sit at room temperature until the next day, when I'm ready to frost the cake.)
In the bowl of your stand mixer fitted with the paddle attachment, beat the cooled brown butter for 1 minute until smooth.
In a separate bowl, combine the powdered sugar, meringue powder and salt. With the mixer on low, add the powdered sugar mixture by spoonfuls, then the vanilla. Increase the speed to medium (#4/6 on a Kitchen Aid mixer), and beat for about 5 minutes until light and fluffy, scraping the bowl several times. The process of browning the butter removes the water from the butter, so this buttercream will be too dry unless you add back in some liquid. So while you're beating the buttercream, add in the milk or cream a tablespoon at a time, until you're happy with the consistency for spreading onto your cake. Turn the speed down to "stir" and mix for 1 minute to pop large air bubbles.
Remove the cooled cakes from the pan. Fill and frost the cake layers with the buttercream, then cover the cake with a thin crumb coat of frosting. Chill for 30 minutes to set the crumb coat.
Add a few tablespoons of sprinkles (such as nonpareils and quins/confetti sprinkles) to the remaining buttercream, and frost the cake all over with a final coat of buttercream.
Before the frosting crusts over, decorate the cake all over with more sprinkles. (I hand placed every sprinkle on my cake using tweezers.)