To poach the chicken, place the chicken pieces in a large saucepan and cover with 2 litres/3½ pints of water.
Add the carrot, onion, leek, garlic, thyme and bay leaf. Bring to a gentle boil then turn down to a simmer and cook for 1¼ hours.
When the meat is tender, remove from the heat. Drain the chicken, reserving 1 litre/1¾ pints of the cooking liquor for making the sauce. Set aside.
To make the sauce, heat a heavy based saucepan over a medium heat, add the butter. When melted, add the flour and keep stirring until it is completely absorbed.
Gradually whisk in the reserved poaching liquor, stirring until all the liquor has been incorporated.
Drain the meat again and discard any remaining liquor along with the carrot, onion, bay leaf, clove and thyme.
Place the meat into the sauce.
Heat a frying pan over a medium heat, add a little more of the butter and the baby onions. Cook until slightly coloured then add the mushrooms and cook for a further two minutes.
Tip the onions and mushrooms into the chicken and sauce.
Whisk the egg yolks and cream together in a bowl then gradually add to the pan, whisking as you go.
Remove from the heat and continue to whisk for a couple of minutes.
Stir in the tarragon and season to taste with salt and pepper. Keep warm until ready to serve.
For the roasted shallots and fresh garden peas, preheat the oven to 200C/400F/Gas 6.
Heat a large griddle pan and drizzle the oil over the cut side of the shallots. Place on the griddle pan cut-side down and cook for 2-3 minutes, or until coloured and charred. Turnover and cook for a few more minutes, then place in the oven for 15-20 minutes.
Meanwhile heat a saucepan of boiling water and season with salt. Once boiling, add the peas and cook for 3-4 minutes then drain.
Heat a frying pan and add the butter and peas, season with salt and pepper and cook gently for a few minutes.
To serve, place a couple of pieces of chicken on each plate and spoon over the sauce. Place the roasted shallots and peas next to the chicken.