Heat 2 tablespoons oil in large skillet over medium heat. Add sausage, garlic, and crushed red pepper; saut� until brown, breaking up with fork, about 6 minutes. Add carrots, onion, and celery; saut� until beginning to brown, about 10 minutes. Mix in tomatoes. Reduce heat, cover, and simmer 15 minutes, stirring occasionally. Add 1-cup broth and wine; simmer uncovered until liquid is slightly reduced, about 15 minutes. Add basil; cover and simmer until vegetables are very tender, adding more broth by 1/4 cupful if liquid evaporates too quickly and stirring occasionally, about 40 minutes. Simmer uncovered until rag� thickens to desired consistency, stirring often, about 10 minutes longer. Season lightly with salt and pepper. (Can be prepared 1 day ahead. Refrigerate uncovered until cold, then cover and keep refrigerated. Rewarm over medium heat before using.)
Cook pasta in large pot of boiling salted water until tender but still firm to bite. Drain, reserving 1/2 cup cooking liquid. Return pasta to pot. Mix in rag�, 1/2 cup cheese, and 2 tablespoons oil. Add cooking liquid by 1/4 cupful to moisten, if desired.
Divide pasta among plates. Serve, passing 1/2 cup cheese separately.