Wipe out the skillet. Add the remaining 1/4 cup of olive oil and heat until shimmering. Add the onion, garlic, crushed red pepper and the remaining 1/2 teaspoon of chopped thyme. Cook over moderate heat, stirring occasionally, until softened and just starting to brown, about 8 minutes. Add the sausage pieces and cook over moderately high heat, breaking up the meat with a spoon, until browned, about 7 minutes. Stir in the rye bread crumbs, mushrooms and 1/2 cup of water and season with salt and pepper. Keep the hash warm over very low heat, stirring occasionally.