Warm the bacon fat or olive oil in a large Dutch oven over medium-high heat. Add onion, celery, carrots, and garlic. Season with salt and pepper.
Cook, stirring occasionally, until the vegetables soften, about 10 minutes.
Stir in the tomato paste, coating the vegetables. Cook until the paste darkens and sticks to the bottom, about 2 minutes.
Pour in chicken broth and lentils. Bring to a boil, then reduce heat to a gentle simmer. Nestle the chicken thighs into the pot, cover, and cook for 1 hour, checking occasionally.
Transfer the chicken thighs to a plate to cool slightly.
Blend about half of the lentil soup in batches to your preferred consistency. Stir the puréed portion back into the pot.
Shred the cooled chicken with two forks and return to the soup. Adjust seasoning with salt and pepper as needed.
Ladle the soup into bowls and sprinkle with grated Parmesan if desired. Serve hot.