- Mash 5 chestnuts with a fork. Set aside.
- In a saucepan over medium heat, soften the carrots in the butter. Add the broth and simmer gently for about 2 minutes. Add the crushed and whole chestnuts. Stirring constantly, reduce until the liquid has evaporated. Remove from the heat and gently add the asparagus and chives. Season with salt and pepper. Cool to room temperature.
- With the rack in the middle position, preheat the oven to 180°C (350°F). Line a 25 x 18-cm (10 x 7-inch) baking dish with parchment paper.
- Butterfly the breasts, slicing them lengthwise down the middle and opening them like a book. Season with salt and pepper. Divide the stuffing among the breasts and close. Place the stuffed breasts in the baking dish.
- In a bowl, beat the egg yolk, mustard and oil with a fork. Brush the chicken with the mixture. Sprinkle with breadcrumbs. Bake until the chicken is no longer pink inside, 25 to 30 minutes.
- Meanwhile, in a small saucepan over medium heat, soften the carrots and garlic in the butter. Deglaze with vinegar and reduce until all liquid has evaporated. Add the broth. Bring to a boil and reduce by half.
- In the blender, purée the sauce until smooth. Season with salt and pepper.
- Spoon the sauce over the chicken and serve with mashed potatoes and green beans.