Put the pearl hairs in a large cooking pot. Pierce the cloves in the entire peeled onions and add them in the cooking pot. Add the sliced carrots, leek and celery, together with the bay leaves, the thyme, the star anise, salt and pepper. Cover with cold water, bring to the boil and simmer for 40 minutes over a low heat.
Preheat the oven to 180 ° C. Cut squares of 9 cm from 6 puff pastry, by cutting 10 strips of 1.2 cm width on the 4 sides. Cut the last 4 dough sheets into 12 strips each. Place 4 strips of each other on each side of the squares and gradually brush everything with the beaten egg. Bake the pastries for 25 minutes in the oven until golden brown.
Remove the guinea fowl from the cooking liquid and sieve this broth. Let the guinea fowl cool. Remove the skin, loosen all the meat and tear the larger pieces in small parts.
Fry the flour in a stew with the melted butter, until everything turns brown. Gradually add 1 l of the sifted broth. Gently bring to the boil and then add the cream, along with the Noilly Prat and the pieces of guinea fowl. Simmer for 2 minutes and season further.
Cut the unpeeled potatoes into fries, rinse them and dry them. Mix them with olive oil, arrange them next to each other on a baking sheet with baking paper and bake for 20 to 25 minutes in the oven until golden brown (preheated at 230 ° C) or bake them twice until golden brown in the frying pan. Mix the grated Parmigiano, the pressed garlic clove, the Fleur de Sel, the pepper and the parsley (well flushed, dried and finely chopped). Season the fries with this mixture (or add it separately).
Brush the mushrooms clean (without water!), Remove the stems and cut the largest specimens into pieces. Fry the finely chopped shallot for 5 minutes on a low heat in a frying pan with the melted butter. Add the mushrooms and lemon juice, put the lid on the pan and let it shrink for 4 minutes. Reduce the juice almost completely, add the cognac and the chopped parsley. Season further.
Carefully cut the hats from the puff pastry cakes, arrange them on the plates and fill them with guinea fowl and sauce. Add the fries and mushrooms.
