In a small saucepan over medium-high heat, melt the butter until the milk solids start to caramelize, turn foamy and a little brown, and smell nutty, about 5 minutes. Pour the brown butter into a bowl and set aside to cool.

Combine the eggs, granulated sugar, orange zest, and vanilla in the bowl of a stand mixer. Whip the mixture on high speed with the whisk attachment until thick and very pale, about 5 minutes. Remove from the mixer and sprinkle in the brown rice flour, almond flour, tapioca starch, baking powder, and salt. Gently fold the ingredients together with a spatula, trying to keep as much air as possible in the batter.

When you don’t see any streaks of flour left, pour in the brown butter and fold to incorporate. Again, try to be gentle to keep as much air as possible. Transfer the bowl to the refrigerator and chill for 1 hour.

Preheat the oven to 425°F (220°C). Grease a madeleine pan with a little melted butter. Fit a pastry bag with a medium plain tip (about ½ inch, or 1.2 cm in diameter), or use a large spoon to move batter into the mold or even a ziplock bag with the tip cut. Fold the top edges of the pastry bag outward, and when the batter is chilled and solid, scoop it with a spatula into the bag. Unfold the edges and twist the bag in your hand to pipe the batter into the madeleine molds, filling them nearly to the top.

Place the pan in the oven, reduce the temperature to 375°F (190°C), and bake for 12 to 15 minutes, or until the madeleines are golden brown, set in the center, and develop a “bump.” Carefully unmold the madeleines and dust with powdered sugar. They are best eaten right away.