Here in Texas, there is always an abundance of fresh tomatoes and avocados so this pasta is a great way to enjoy both of these wonderful ingredients. The avocado adds a wonderful richness to this "uncooked" sauce.
In a large pot, bring 4 quarts water to a rapid boil over high heat. Stir in 1 tablespoons salt. Drop in pasta and cook, stirring frequently, until tender but still firm to the bite.
Meanwhile, scoop the avocado pulp into a large bowl. Using a fork, mash the avocado coarsely, leaving a few small chunks. Add the lime or lemon juice, onion, tomato, chopped basil, and salt and pepper to taste. Mix gently.
Drain the pasta, transfer to the bowl with the avocado mixture, and toss well. Garnish with basil sprigs and serve immediately. Pass the cheese at the table for sprinkling over individual servings.