Bring a large pot of salted water to a boil. Meanwhile, whirl garlic in a food processor to mince. Scoop soft-ripe avocados from peels into food processor. Add 1 cup basil, the parsley, chives, 1 cup parmesan, the lemon juice, remaining 1 tsp. salt, and the pepper; whirl until smooth.
Cook pasta according to package directions until just tender. Meanwhile, dice the firm-ripe avocado. Reserve 1 1/2 cups pasta water, then drain pasta and return to pot. Add avocado sauce and as much pasta water as you like (to loosen texture); toss gently to combine.
Transfer pasta to a serving bowl and gently stir in diced avocado. Grate a little more parmesan on top. Thinly slice remaining 1 tbsp. basil and scatter over pasta. Serve immediately.