Lentil Soup With Winter Vegetables

Name: Lentil Soup With Winter Vegetables
Ingredients: 8 ounces lentils du Puy; 1/2 cup (1 stick) unsalted butter; 1 cup carrots, finely chopped; 1 cup onion, finely chopped; 2/3 cup celeriac, finely chopped; 2/3 cup leeks, finely chopped; 2/3 cup celery, finely chopped; 2 sprigs fresh thyme; 4 cloves garlic; 10 black peppercorns; 5 cups homemade or store-bought low-sodium chicken stock, heated, plus more as needed; Coarse salt and freshly ground pepper; 20 baby spinach leaves; Extra-virgin olive oil, for drizzling; 4 large eggs; 1/3 cup white vinegar