Preheat oven to 350°. Put mushrooms in a small bowl and pour 2 cups simmering water over them. Stir and let mushrooms sit 15 minutes. In another small bowl, pour milk over bread cubes. Squeeze bread with your fingers to soften completely.
Break the meats into small chunks, then toss together in a large bowl. Add shallots, parsley, eggs, salt, and pepper.
Use a slotted spoon to remove mushrooms from water; reserve liquid. Chop mushrooms fine and add about three-quarters to meat mixture; set remaining mushrooms aside. Gently squeeze bread to remove excess milk and add it to meat mixture; discard milk. Using your hands, gently mix bread, meat, and seasonings until blended. Form mixture into a ball, transfer to a large, heavy-bottomed metal roasting pan, and pat into a rounded loaf.
Sprinkle carrot, celery, and remaining mushrooms in bottom of pan. Pour in 3/4 cup each of mushroom soaking liquid (strain out any sediment, if necessary) and chicken stock. Bake until outside of loaf is browned and interior measures 180° on a meat thermometer, about 90 minutes.
Transfer meat loaf to a platter. Set roasting pan over 2 stovetop burners on high heat. Pour in wine and use a wooden spoon to loosen vegetables and browned bits. Add remaining mushroom liquid and chicken stock and simmer until liquid is reduced by half. Reduce heat to medium and whisk in butter; sauce will thicken slightly. Transfer sauce (with vegetable bits) to a gravy boat and serve alongside meat loaf.
Note: Nutritional analysis is per serving.