In food processor, pulse together garlic, onion, carrot and mushrooms until finely chopped.
Heat oil in large skillet on medium. Add chopped vegetables, oregano, red pepper flakes and ¼ teaspoon each salt and pepper and cook, covered, stirring occasionally, until tender, 4 to 5 minutes. Add beef and cook, breaking up with wooden spoon, until browned, 10 to 12 minutes.
Stir in tomato paste and cook 1 minute. Add tomatoes (and their juices), crushing with your hands as you add to skillet. Add lentils, 2 cups water and ¼ teaspoon each salt and pepper; simmer until lentils are tender, 15 to 20 minutes.
Meanwhile, cook pasta per package directions. Drain and toss with sauce. Top with pine nuts and Pecorino, if desired.
