How to Make It
Preheat the oven to 450°. On a rimmed baking sheet, toss the rye bread pieces with 2 tablespoons of the olive oil and season with salt and pepper. Bake for about 5 minutes, until the crumbs are light golden but not completely dry.
In a large skillet, heat 2 tablespoons of the olive oil until shimmering. Add the mushrooms, baking powder and 1/2 teaspoon of the thyme and season with salt and pepper. Cook over moderately high heat, stirring occasionally, until the mushrooms are tender and browned, about 8 minutes. Transfer the mushrooms to a plate.
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.
Meanwhile, bring a very large, deep skillet of water to a simmer over moderate heat. Crack the eggs one at a time into a small bowl and carefully add to the simmering water. Poach the eggs over moderate heat until the whites are set but the yolks are still slightly runny, about 5 minutes.
Using a slotted spoon, carefully lift the poached eggs out of the water; blot them dry with paper towels. Spoon the hash onto 4 plates and top with the eggs. Garnish the hash with fleur de sel and snipped chives and serve with hot sauce.