Melt butter in a light-colored saucepan over medium-low heat. Keep it moving and cook for about 5 to 8 minutes, until the foam subsides and you see golden-brown flecks on the bottom. It should have a fragrant nutty aroma.
Pour it into a bowl, scraping in any flecks, and let it cool for 10 minutes.
Sift the almond flour, all-purpose flour, and powdered sugar together into a large bowl.
Stir in the egg whites, vanilla extract, almond extract, the cooled browned butter, and salt. Mix just until smooth.
Cover and refrigerate the batter for 2 hours.
Preheat oven to 350°F.
Generously grease a mini muffin tin with softened butter.
Scrape the batter into a piping bag and pipe into the muffin cups, filling each no more than ¾ full.
Sprinkle the sliced almonds over the tops.
Bake for 12 to 15 minutes, until the batter no longer looks wet anywhere and the edges are golden brown. Don't overbake.
Cool in the pan for 10 minutes, then turn them out onto a rack.
Notes
Storage: Airtight at room temperature up to 3 days, or freeze up to 2 months. Batter keeps 2 days in the fridge.
Don't overbake, or they'll dry out. Pull them the moment the centers set.
Grease the pan generously for clean edges and easy release.