Place the guinea fowl in a large pot. Stick the cloves into the whole peeled onions and add them to the pot. Add the chopped carrots, leeks, and celery, along with the bay leaves, thyme, star anise, salt, and pepper. Cover with cold water, bring to a boil, and let simmer for 40 minutes on low heat.
Preheat the oven to 180°C. Cut 9 cm squares from 6 sheets of puff pastry by cutting 10 strips of 1.2 cm width from all 4 sides. Cut the last 4 sheets into 12 strips each. Place 4 strips in a staggered manner on each side of the squares and gradually brush everything with the beaten egg. Bake the pastries in the oven for 25 minutes until golden brown.
Remove the guinea fowl from the cooking liquid and strain this broth. Let the guinea fowl cool. Remove the skin, pull all the meat off, and tear the larger pieces into smaller parts.
Sauté the flour in a stew pot with the melted butter until everything turns brown. Gradually add 1 liter of the strained broth while whisking. Gently bring to a boil and then add the cream, along with the pieces of guinea fowl. Let 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 side by side on a baking sheet lined with parchment paper, and bake them in the oven for 20 to 25 minutes until golden brown (preheated to 230°C) or fry them in two batches until golden brown. Mix the grated Parmigiano, crushed garlic clove, fleur de sel, pepper, and parsley (well rinsed, dried, and finely chopped). Season the fries with this mixture (or serve it separately).
Brush the mushrooms clean (without water!), remove the stems, and cut the largest ones into pieces. Sauté the finely chopped shallot for 5 minutes on low heat in a frying pan with the melted butter. Add the mushrooms and lemon juice, cover the pan, and let wilt for 4 minutes. Let the juice reduce almost completely, add the cognac and chopped parsley. Season further.
Carefully cut the tops off the puff pastry cases, arrange them on the plates, and fill them with guinea fowl and sauce. Serve with the fries and mushrooms.
