To make the potato case: Line a tall 9-inch springform cake pan with baking paper, and grease the inside of the paper.
Peel and grate the potatoes on a coarse grater. Put them into a clean tea towel and squeeze out as much liquid as you can, then transfer them to a large mixing bowl. Add Parmesan, salt and the melted butter.
Pour the potatoes into the cake pan and press it up the sides of the pan and across the base, to make a firm case for the quiche. Put the pan into the freezer for 2 to 3 hours, or until frozen solid.
Preheat the oven to 350 degrees. Bake the potato case for 30 minutes then carefully remove it from the pan and put it onto a wire rack on top of a baking tray. Brush the inside and outside with egg wash, making sure to seal the inside with the egg wash. Bake the case again for 15 minutes.
To make the filling: Fry the bacon until crispy and set aside to cool. In a large mixing bowl, whisk the eggs and egg yolks together, then beat in the cream cheese, crème fraîche and heavy cream. Once the mixture is smooth, add in the grated Gruyere, chopped herbs, salt and pepper.
Take the potato case off the wire rack and put it onto a rimmed baking tray. Pour the filling into the case and bake for 25 minutes, then let it cool for 30 minutes before serving.