Finely chop garlic in food processor. Add all remaining ingredients; blend until smooth and pale green. Season sauce to taste with salt and pepper. Transfer sauce to serving bowl. Cover and refrigerate until cold. (Can be prepared 2 days ahead. Keep refrigerated.)
Fill 2 large bowls with ice water; set aside. Bring very large pot of water to boil over high heat. Add 1/2 cup coarse salt to boiling water; return to boil. Rub shrimp in another large bowl with remaining 1/4 cup salt. Add shrimp to boiling salted water and cook just until cooked through, about 3 minutes (water will not return to boil). Immediately drain shrimp in colander, then quickly divide between prepared bowls of ice water; cool completely. Drain well. Peel and devein shrimp, leaving tails intact. Arrange shrimp on platter. (Can be made 4 hours ahead. Cover and refrigerate.) Serve shrimp with sauce.
