In a saucepan, heat unsalted butter over medium-low heat until it turns golden brown and emits a nutty aroma. Transfer to a bowl to cool.
Whisk together flour, cocoa powder, baking powder, and salt in a separate bowl.
In a large bowl, beat sugar, eggs, and vanilla until pale and shiny. Fold in the dry ingredients.
Gently fold in the cooled brown butter, then chill the batter for 30 minutes.
Preheat oven to 350°F. Brush melted butter into each well of the madeleine pan.
Spoon about 1 ½ tablespoons of batter into each well and bake for 9-11 minutes until domed.
Cool in the pan for 5 minutes, then transfer to a wire rack to cool completely.
For the chocolate dip, melt chocolate chips and coconut oil together. Dip each madeleine and sprinkle with flaky sea salt.
Chill to set the chocolate, then serve and enjoy these heavenly treats!