Drop the pancetta into the food processor bowl, and pulse several times, to chop the meat into small bits.
Scrape all the chopped pancetta right into a heavy 5- to 6-quart saucepan.
Put the onion, carrot, celery, and sage leaves into the empty food processor bowl, and mince together into a fine-textured paste or pestata.
Put the butter and olive oil into the saucepan with the minced pancetta, and set over med-high heat.
Cook, stirring, as the butter melts and the fat starts to render.
When the pancetta is sizzling, scrape in the vegetable pestata, and stir it around the pan until it has dried and begins to stick, 4 minutes or so.
Clear a space on the pan bottom and drop in the tomato paste, toast it in the hot spot for a minute, then stir together with the pestata.
Increase the heat, pour in the wine, and cook, stirring, until the wine has almost completely evaporated.
Pour in 8 cups of hot water and the tablespoon of salt, stir well, and heat to a boil (add all 10 cups of hot water if you want to serve the rice and lentils as a thick soup rather than a denser riso).
Cover the pan, and decrease the heat slightly, to keep the water at a moderate boil, and let it bubble for 20 minutes or so, to deepen the flavors.
Stir in the lentils, return to a gentle boil, and cook, partially covered, stirring occasionally, until the lentils just start to soften, 10-15 minutes.
Stir in the rice, return to a bubbling simmer, and cook, cover ajar, until the rice is al dente, 13 minutes or so.
If the dish is thickening more than you like, lower the heat and cover the pan completely.
If it seems too thin, remove the cover and cook at a faster boil.
When the rice and lentils are fully cooked, turn off the heat.
Stir in the scallions and grated cheese.
Serve in warm bowls, passing more cheese at the table.