In a soup pot, melt 2 tbsp of the butter with all the oil. Add onion and pancetta, set over medium-high heat, and cook uncovered, stirring, until the onion turns a deep gold.
Add carrot and celery and cook 2 to 3 minutes at lively heat, stirring occasionally.
Add the tomatoes with their juice. Lower heat so they bubble gently but steadily and cook about 25 minutes, stirring now and then.
Rinse the lentils in cold water and drain. Add them to the pot and stir to coat, then add the broth, a pinch of salt, and a few grindings of pepper. Cover and simmer gently, stirring from time to time, until the lentils are tender, about 45 minutes (taste to judge). Add more broth, or water if not using homemade broth, if needed.
When the lentils are tender, swirl in the remaining 1 tbsp butter and the Parmesan before turning off the heat. Taste and adjust salt and pepper, and serve with additional grated Parmesan.