Cut the unsalted butter into cubes and melt it in a small saucepan, over a medium to high heat. Once melted, you want to keep heating it, stirring occasionally with a wooden spoon, until brown specks appear and you can feel that nutty aroma. Be careful not to burn the butter, but also don't "underbrown" it - if you know what I mean.
Once your butter is browned, transfer it in a bowl and let it cool a little bit, but it still has to be melted.
In a bowl of a stand mixer, using a whisk attachment, beat together eggs, vanilla extract and sugar on high speed for 5-7 minutes. The mixture should be paler, doubled in size, thick and fluffy. It should also fall in ribbons when you pick up the whisk.
In another bowl, sift together flour, baking powder and salt. Using a wooden spoon or a silicone spatula, fold it in the egg mixture. Be gentle so you don't knock out all the air out.
Add a couple of tablespoons of the batter into melted brown butter and stir to combine it throughly. There shouldn't be any traces of butter visible. Transfer it back to the remaining butter and fold until everything is well incorporated.
Cover the batter and chill it in the fridge for about 30-45 minutes. This will ensure that characteristic madeleine bump.
Preheat your fan oven to 165°C (325°F) or 175°C if you’re not using a fan oven. Thoroughly grease your madeleine pan so the cookies won't stick.
Place approx. one tablespoon of chilled batter into the deepest part of the pan. There is no need to flatten or spread it.
Bake in a preheated oven, on the middle rack, for about 10-12 minutes. Madeleines should puff in the middle and be crispy on he outside.
When they're done, leave them to cool in the pan for 2 minutes and then transfer them to a cooling rack to cool completely.
Caramelised White Chocolate
Preheat your fan oven to 120°C (250°F). Place chopped good quality white chocolate (at least 29% cocoa solids) in a medium sized non stick rectangular pan. For easier handling later, you can line your pan with greaseproof paper.
Heat/melt chocolate in intervals of 10 minutes. After every 10 minutes, take out the pan and using a silicone spatula scrape the chocolate and spread it in a thin creamy layer.
You do this to prevent chocolate burning. I usually do it for a total of an hour, but just do it until white chocolate turns caramely colour. More like a peanut butter. The longer you heat it, the darker it will be. Just don't burn it. :)
When you're satisfied with your chocolate it's time to chill it. Spread the chocolate evenly and leave it to cool to room temperature in the same pan. After that, put it in the freezer until it becomes rock hard. Since the chocolate isn't tempered, this is the only way you will prevent it from melting while you break it into chunks to save for later.
Place it in a jar and keep it in the fridge. For dipping madeleines, melt 100 g of chunks over a bain marie.