Preheat the oven to 200C/180C (fan)Gas 6. Line a large baking tray with baking parchment and set aside.
Pour 125ml/4fl oz water into a medium non-stick saucepan set over a low heat and add the butter. Sift the flour into a bowl and stir in the sugar.
Melt the butter slowly in the water, then increase the heat and bring it to the boil. Immediately tip all the flour into the bubbling water and stir vigorously with a wooden spoon until the mixture forms a thick, smooth paste and leaves the sides of the pan. Remove from the heat.
Continue to stir for 2–3 minutes, until the paste has cooled a little. Add the beaten eggs, just a little at a time to the choux paste, beating very well between each addition. The paste will become softer and more glossy as the eggs are added. (You can use an electric mixer to incorporate the eggs if you prefer.)
Spoon 18 heaped teaspoons of the choux paste onto the baking tray. Space them well apart to allow for rising. If you dip the teaspoon into a jug of hot water every now and then, it will help prevent the paste sticking. Use a second teaspoon to push the mixture off the spoon and only the tray.
Bake for 25-30 minutes, or until the buns are well risen and golden-brown.
Remove the tray from the oven. Holding each bun with a clean, dry tea towel, turn upside down on a wire rack and make a 1cm/½in hole in the base by twisting the tip of a knife or poking with a skewer to allow the steam to escape. Leave to cool.
While the profiteroles are cooling, make the white chocolate sauce. Place the white chocolate and 150ml/5fl oz of the cream in a small non-stick saucepan and melt over a very low heat, stirring regularly until smooth. Do not allow to overheat.
Remove from the heat and leave to cool. The sauce will look quite yellow at this point, so pour into a jug and leave to cool for at least 30 minutes before using and it will become whiter.
Make the dark chocolate sauce in the same way, using the remaining 100ml/3½fl oz of cream, and set aside.
Make the vanilla cream filling for the profiteroles by whipping the cream with the sugar and vanilla until soft peaks form when the whisk is removed from the bowl. Spoon into a large piping bag fitted with a 1cm/½in plain nozzle. Pipe the cream into each choux bun through the base. Alternatively, split the buns and spoon the cream inside. Pile up on a serving plate and dust with sifted icing sugar. (The buns can be covered and popped into the fridge for up to six hours before serving if you like.)
When you are ready to serve, stir three tablespoons of just-boiled water into the dark chocolate sauce, one tablespoon at a time, and heat gently, stirring until smooth and hot. Pour the dark chocolate sauce over the profiteroles. Drizzle the white chocolate over the top and serve immediately. (If the white chocolate sauce has become too firm to pour, stir in a couple of tablespoons of just-boiled water first.)