Bring a large saucepan filled with water to a boil. Cook beans for about 3 minutes (a minute or two longer if not using haricot verts) or until crisp-tender. Plunge beans into ice water to stop cooking and drain well.
Put bread in food processor and pulse until coarse crumbs form. Remove to a bowl and mix in parsley, 1 Tbsp olive oil, minced garlic, and 2 Tbsp grated parmesan cheese.
Pre-heat broiler to high.
Heat remaining 2 Tbsp olive oil in a 10-inch cast-iron skillet over medium heat. Add onion and mushrooms and sauté for about 8 minutes. Stir in flour and cook for 1 minute.
Stir stock, milk, and sherry into the skillet and bring to a simmer. Cook for about 3 minutes.
Remove skillet from heat and stir in remaining 4 Tbsp parmesan, salt, pepper, and nutmeg. Add beans to pan and toss to coat in mushroom/onion mixture.
Sprinkle skillet with breadcrumb mixture and place under broiler for 2 minutes.