Toast garlic in a small, dry skillet over medium heat, shaking pan occasionally until the garlic is softened and spotty brown. When cool, remove and discard the skins.
In a food processor, pulse the toasted garlic, olives, shallot, basil, parsley, cheese, olive oil, and lemon juice, stopping when necessary to scrape the sides of the bow. Add salt and pepper to taste (I added none).
Cook spaghetti until al dente. Drain, toss with pesto mixture. Serve immediately with lemon wedges and extra cheese.