Put the chicken into a large pot. Pour over the wine and chicken stock. Add water if necessary so the chicken is covered. Add the carrots, celery and onions and season with salt and pepper. Put over medium-high heat and bring it to a boil. Turn off the heat and cover the pan. Let is sit for 30 minutes so the chicken finishes cooking. Remove the chicken from the pot and keep warm, reserving the poaching liquid.
Strain 2 cups of the poaching liquid into a medium pot over high heat. Bring to a boil and reduce the liquid by half. Whisk in the cream and reduce by half again. Cover and keep warm.
Strain 1 cup of the poaching liquid into another medium pot over high heat. Bring to a boil. Reduce the heat to a simmer and add the asparagus tips. Cook until they are just tender and bright green, about 1 minute. Drain and keep warm.
When ready to serve, toast the bread and spread it with 2 of the tablespoons butter. Whisk the remaining 2 tablespoons butter into the cream sauce. Place the toast onto a plate, top with a chicken quarter and ladle some cream sauce over it. Garnish with chives. Serve with asparagus tips and Crispy Potatoes au Gratin.
Crispy Potatoes au Gratin:
Preheat the oven to 350 degrees F. Grease 4 shallow ovenproof ramekins with the softened butter.
Slice the potatoes as thinly as possible. Fan a layer of potatoes into the ramekins. Season with salt and pepper and sprinkle over 2 tablespoons cheese. Add another layer of potato and season with salt and pepper and sprinkle over 2 more tablespoons cheese. Put the ramekins onto a baking sheet and pour the cream over the potatoes in each dish. Dot with the cold butter. Bake until brown and crispy, 20 to 25 minutes.