In a medium saucepan, combine the reserved capon necks and gizzards with the shiitake stems, dried porcini and chicken stock and bring to a boil. Reduce the heat to moderate and simmer until reduced to 2 cups, about 1 hour and 15 minutes. Strain the stock into a heatproof bowl, pressing hard on the solids. Season lightly with salt and pepper.
Meanwhile, preheat the oven to 350°. In a food processor, puree 1 1/2 sticks of the butter with the chives, parsley, garlic, lemon juice and 1/2 teaspoon of salt. Using your fingers and starting at the neck, gently loosen the capon skin from the breast and legs. Rub all but 2 tablespoons of the herb butter under the skin of the birds; press gently on the skin to evenly distribute the butter.
Set the capons on 2 racks in 1 large or 2 medium roasting pans. Rub with 1 tablespoon of the olive oil and season generously with salt and pepper. Roast the capons for about 1 hour and 40 minutes, or until golden and an instant-read thermometer inserted in the inner thigh of each capon registers 170°. The juices in the cavity will still be slightly pink. Transfer the birds to a carving board, cover loosely with foil and let stand for at least 10 minutes or for up to 30 minutes.
Meanwhile, in a large, deep nonstick skillet, heat the remaining 2 tablespoons of olive oil. Add the shiitakes, season with salt and pepper and cook over moderately high heat until browned, 8 minutes. Add the remaining 2 tablespoons of unsalted butter and cook, stirring, until melted. Add the shallots and cook, stirring occasionally, until golden, about 6 minutes.
Add the flour to the mushrooms and cook, stirring, for 2 minutes. Add the Madeira and bring to a boil, stirring constantly, for 1 minute longer. Add the strained stock and bring to a boil, stirring. Reduce the heat to moderately low and simmer for 30 minutes. Whisk in the remaining 2 tablespoons of herb butter and season with salt and pepper; keep warm.
Pour the juices from the roasting pan into a heatproof bowl and skim off the fat. Set the roasting pan over 2 burners on high heat, add the water and scrape to release any browned bits from the bottom; add to the pan juices in the bowl. Strain the pan juices into the Madeira sauce, along with any accumulated drippings from the capons; stir and season with salt and pepper. Carve the birds and serve them with the Madeira sauce on the side.
Make the stock and butter the capons up to 1 day ahead. Bring the capons to room temperature before roasting; finish the sauce while the birds rest before carving.
A full-bodied fruity Chardonnay or an earthy-fruity Pinot Noir from Burgundy echoes similar notes in the capons and their rich mushroomy sauce.