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Place lentils in a large bowl. Add 2 1/2 cups water; let soak for 12 hours.
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Melt 6 tablespoons butter in a large Dutch oven over medium heat. Add carrots, onion, celeriac, leeks, and celery. Cook, stirring, until vegetables have softened and onions are translucent, 2 to 3 minutes.
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Drain and rinse lentils; add to Dutch oven. Cook, stirring to combine, 2 to 3 minutes. Place thyme, garlic, and peppercorns in a piece of cheesecloth; tie with kitchen twine to enclose. Add to skillet along with chicken stock. Place a parchment paper round over liquid and cook for 10 minutes. Season with salt and continue cooking until lentils are tender, about 5 minutes more.
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Transfer 1/3 of the soup mixture to the jar of a blender; blend until smooth, adding chicken stock, if necessary. Transfer blended soup back into Dutch-oven with remaining soup; stir to combine. Add spinach leaves, remaining 2 tablespoons butter, and drizzle with olive oil. Keep warm.
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Fill a large pot with water and add vinegar; bring to a boil over medium-high heat. Gently crack eggs into pot, one at a time, and reduce heat to medium. Cook eggs for 3 minutes; using a slotted spoon, transfer cooked eggs to a paper towel-lined plate.
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Serve soup immediately topped with a poached egg.