Place chicken into a large saucepan and cover with 2.2 litres of water.
Add carrot, onion, garlic, and thyme, then simmer for 1.25 hours.
Strain the liquid into a bowl and reserve 1 litre of the liquor for the sauce.
Pick the chicken meat from the bones and discard the vegetables and herbs.
Melt butter in a saucepan and stir in flour to create a roux.
Gradually whisk in the reserved 1 litre of chicken liquor until smooth.
Remove the pan from the heat and allow it to cool for one minute.
Whisk egg yolks and cream together then slowly stir into the sauce.
Fry baby onions and mushrooms in a separate pan with butter until slightly coloured.
Stir the vegetables, tarragon, chicken, and seasoning into the sauce off the heat.
Heat butter and oil in a pan until foaming for the potatoes.
Fry diced potatoes for 10 to 12 minutes until golden and tender.
Add garlic and parsley to the potatoes and season with salt and pepper.
Serve the chicken and sauce on plates alongside the fried potatoes.