Melt the dark chocolate in a heatproof bowl over a saucepan of barely simmering water. Stir until smooth, remove from heat and let cool slightly.
in a bowl, beat the softened butter with the two sugars with a wooden spoon for about 1 minute until the mixture is smooth and not frothy (you can also use an electric whisk). Gradually add the beaten eggs, mixing well between each addition. Add the vanilla and mix again.
Pour the melted chocolate into the mixture and stir until combined. Scrape down the sides of the bowl with a spatula. Sift the flour, cocoa, baking soda, baking powder and salt. Mix again until well blended. Line two large baking sheets with parchment paper.
Stir three-quarters of the white chocolate into the cookie mixture. Using two dessert spoons, shape the mixture into 20-22 roughly shaped but evenly sized balls (each about 2oz/55g) and place on a lined baking sheet. Press remaining white chocolate into cookie dough balls. Cover and refrigerate for 1 hour until firm, then roll each cookie ball in your hands until you have a smooth, firm ball. Cover and refrigerate again for 4 hours or overnight.
Preheat the oven to 170C/150C Hot Fan/Gas 3.
Place 6-8 balls of dough on each baking sheet, leaving plenty of space between each as the cookies will spread while baking. Bake for 12 minutes then remove from the oven.
You now have two choices: either bang the bottom of the tray on the work surface to deflate the cookies, or take a 10cm cookie cutter and place a cookie inside the mold, still on the baking tray, and swirl it around to make an even, crisp round cookie.
Repeat with all cookies. Return the cookies to the oven for an additional 2-4 minutes until firm but not crispy (baking for an additional 2 minutes will result in a softer cookie and after 4 minutes they will be crispier – depending on your preference).
The cookies are soft when baked but will firm up as they cool. Continue baking the cookies in batches. Cool on baking sheets until firm, then transfer to wire rack to cool completely. Store in airtight containers between layers of parchment paper.