For the sabayon, place a glass bowl over a saucepan of barely simmering water. Add the egg yolks and sugar and whisk continuously, until pale and light.
Add the champagne and continue whisking until pale and fluffy. (The foam must be sufficiently thickened so that it won't break down when it cools.) Remove from the heat and cool until tepid.
For the berry gratin, place all of the berries and the sugar into a bowl and mix together. Then divide evenly among four ovenproof bowls.
Add the basil to the sabayon, mix well, then spoon the sabayon over the berries.
Using a mini-blowtorch, heat the top of the sabayon until it turns golden-brown, then serve immediately.