Place chicken pieces in a large saucepan and cover with 2 litres of water.
Add carrot, onion, leek, garlic, thyme, and bay leaf then simmer for 1.25 hours.
Drain the chicken and reserve 1 litre of the poaching liquor.
Melt 25g butter in a saucepan and stir in the flour until absorbed.
Gradually whisk in the reserved chicken liquor until incorporated.
Discard the aromatics and place the poached meat into the sauce.
Sauté baby onions and mushrooms in butter then stir into the chicken sauce.
Whisk egg yolks and cream together then gradually whisk into the pan.
Stir in the tarragon and season with salt and pepper.
Preheat the oven to 200°C.
Grill shallots cut-side down for 3 minutes then roast in the oven for 20 minutes.
Boil the peas for 4 minutes then drain.
Sauté the peas with 50g butter and serve alongside the chicken and shallots.