How to Make It
Preheat the oven to 350°. In a heatproof bowl, soak the porcini in the boiling water until softened, about 20 minutes. Rub the porcini to remove any grit, then lift them out of the soaking liquid and finely chop. Reserve the liquid.
Melt 2 tablespoons of the butter in a medium skillet. Add the carrot and half of the onion and cook over low heat, stirring occasionally, until softened, about 8 minutes. Add the liver, season with salt and pepper and cook until it is no longer pink on the outside, about 2 minutes. Stir in the chopped porcini. Transfer the mixture to a medium bowl and let cool completely. Add the ground veal, ham, eggs, 1 teaspoon of salt and 1/2 teaspoon of pepper. Mix with your hands until blended.
Spoon the stuffing into the cavity of the capon and tie the legs together. Transfer the capon to a roasting pan and rub all over with the tablespoon of softened butter. Pour the wine over the capon and season it with salt and pepper. Add the remaining half of the onion to the pan and roast the capon for about 45 minutes, or until the wine has almost evaporated, basting once halfway through. Add the chicken stock to the pan and baste again. Roast for about 2 hours longer, or until an instant-read thermometer inserted in the inner thigh registers 160°.
Transfer the capon to a carving board and let rest for 10 minutes. Meanwhile, strain the pan juices into a small saucepan. Set the roasting pan over 2 burners, add the water and simmer over moderate heat for 3 minutes, scraping up the browned bits from the bottom of the pan. Add the reserved mushroom soaking liquid, stopping before you reach the grit at the bottom. Add the pan juices to the saucepan; season with salt and pepper and keep warm over low heat. Spoon the stuffing into a warmed bowl. Carve the capon and serve with the stuffing and pan juices.