For the hazelnut gnocchi, in a bowl, mix together the mashed potato, beaten egg, hazelnut oil and chopped hazelnuts until well combined.
Gradually add the pasta flour, a few tablespoonfuls at a time, mixing well after each addition so that the flour is fully incorporated into the mixture.
Season, to taste, with salt and freshly ground black pepper.
Using your hands, bring the mixture together as a dough. Turn the dough out onto a lightly floured work surface, knead briefly, and roll to a 2cm/¾in thickness.
Cut the gnocchi into 1cm/½in pieces, then squash each piece lightly with a fork, making an indentation in the top.
Bring a pan of salted water to the boil. Carefully lower the gnocchi into the water in batches and return the water to the boil. Once the gnocchi have risen to the surface of the water, cook for 2-3 minutes, then drain well and refresh in cold water.
Heat the tablespoon of olive oil in a frying pan over a high heat, add the boiled gnocchi and fry for a further 2-3 minutes, stirring occasionally, until golden-brown on all sides. Set aside and keep warm.
For the sauce, in a separate frying pan, heat the olive oil over a medium heat, add the girolle mushrooms and the thyme leaves and season, to taste, with salt and freshly ground black pepper. Fry for 2-3 minutes, or until the mushrooms have softened and are golden-brown.
Add the shallots and fry for a further 2-3 minutes.
Add the hot stock and butter and bring the mixture to a simmer. Simmer for 1-2 minutes, or until the butter has melted and the volume of liquid has reduced slightly. Remove from the heat and set aside.
For the eggs, in a separate frying pan, heat the olive oil over a medium heat, crack in the duck eggs and fry for 2-3 minutes, or until crisp and golden-brown underneath and soft on top, or until cooked to your liking.
To serve, divide the hazelnut gnocchi among four serving bowls. Spoon the sauce on top. Place one of the duck eggs on top of each serving of gnocchi. Sprinkle over the reserved whole roasted hazelnuts. Garnish with the parsley shoots, if using, and grated parmesan.