In a saucepan, melt the butter over medium heat. The butter will separate into melted golden fat and white, grainy milk solids. Cook until the milk solids turn golden brown and the mixture smells like toasted nuts, making "beurre noisette". Strain through a fine sieve to remove the milk solids.
Sift the dry ingredients together and place in a mixer fitted with a whip attachment. Add the egg whites, applesauce and strained beurre noisette and mix. The batter can be made to this point up to 3 days in advance. Keep refrigerated. When ready to bake, butter tiny rectangular or round tart molds and cut the blackberries in half, top to bottom. Preheat the oven to 375 degrees F.
Transfer the batter to a pastry bag fitted with a large plain tip and pipe into the prepared molds. Place 1/2 of a blackberry in the center of each mold, cut side down. Bake for 10 minutes, until golden brown. Let cool in the molds, then gently lift the cakes out. Sprinkle with powdered sugar and serve.