Heat the oil in a heavy stockpot over medium heat. Add the onion, celery, carrot, and bay leaves. Cook until the vegetables soften and begin to caramelize, about 10 minutes.
Stir in the tomato paste, salt, and garlic. Cook until fragrant, about 1 minute.
Pour in the broth and add the lentils. Bring to a boil, then reduce heat to maintain a gentle simmer. Cover partially and cook until the lentils are tender, about 25 minutes.
Fold in the spinach, parsley, vinegar, mustard, and pepper. Simmer uncovered until the greens are wilted and flavors have melded, about 15 minutes.
Remove the bay leaves. Ladle into bowls and top with grated cheese.
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