Melt the butter in a saucepan over medium heat until golden brown and fragrant. Remove from heat and cool for 5 minutes.
Preheat oven to 350°F. Grease a madeleine pan well.
Place toasted hazelnuts in a food processor and pulse until finely ground.
In a bowl, whisk together flour, ground hazelnuts, and salt.
In a separate bowl, whisk eggs and sugar until lighter in color and slightly thickened.
Add vanilla extract, hazelnut extract, and browned butter to the egg mixture. Whisk until combined.
Fold the dry ingredients into the wet ingredients until no dry streaks remain. Do not overmix.
Cover the bowl and refrigerate for at least 2 hours.
Spoon or pipe batter into each madeleine cavity, filling about 3/4 full.
Bake for 10 to 12 minutes or until edges are golden and centers spring back lightly.
Let cool in the pan for 2 minutes, then transfer to a wire rack.
Melt chocolate in a microwave-safe bowl in 20-second intervals, stirring between each interval.
Dip one side of each cooled madeleine in melted chocolate and sprinkle with chopped hazelnuts.
Let chocolate set before serving.