Bring a large Dutch oven filled with water to a boil over high. Fill a large bowl with ice water. Place basil leaves in a metal strainer. Place strainer in pan, using tongs to quickly submerge all basil leaves; cook 5 seconds or just until leaves turn bright green. Carefully remove strainer with leaves from pan; drain. Immediately plunge strainer with leaves in ice water bath. Let stand 10 seconds. Remove basil; drain well. Spread basil leaves on a clean, dry dish towel; gently blot dry with another towel.
Place basil, sunflower seeds, oil, salt, and garlic in a food processor; process until smooth. Add cheese; process until blended.