Preheat the oven to 350 degrees F.
In a medium-sized bowl, mix together the veal, pork sausage, ham, and carrot until well combined. Season the stuffing, to taste, with salt and pepper.
Lay the deboned chicken skin-side down on a work surface with the neck end closest to you. Being careful to leave a 1-inch margin on all sides, spread the stuffing evenly onto the chicken. Lay the egg quarters on top. If your butcher left any leg flesh on the sides of the chicken, roll this flesh in over the stuffing. Begin to roll the chicken from the side closest to you the end farthest away, tucking the egg quarters in as you roll so that they are dispersed all along the east-west plane of the chicken. This way, when the roll is cut, every portion will include an egg. When you are finished, the chicken will look like a long, plump salami.
Place the chicken roll in a lightly oiled roasting pan, drizzle with the remaining olive oil and place in the oven. Cook until golden brown, about 30 minutes. Remove from the oven, allow to cool for a few minutes, then cut into 1 1/2-inch slices. Serve immediately.
Recipe courtesy of Mario Batali