Heat the olive oil in a large Dutch oven or deep skillet over medium heat. Add the onion, carrot, and a pinch of salt and cook 6 minutes, until softened.
Stir in the garlic, tomato paste, and oregano. Cook 1 minute, until the paste darkens and smells sweet.
Add the rinsed lentils, crushed tomatoes, broth, and 1/2 tsp salt. Bring to a boil, then lower to a simmer and cook 8 minutes.
Stir in the broken spaghetti and keep the pot at a gentle boil for 12 minutes, stirring every 2 minutes so the pasta does not stick. If it looks dry, add water 1/4 cup at a time.
When the pasta is tender and the sauce is thick, pull the pot off the heat. Stir in the Parmesan and a few grinds of black pepper, rest 5 minutes so the sauce tightens, then serve with more Parmesan.