Place eggs, cheese, milk, onion, salt, if desired, basil, juice, mustard, sauce and pepper in 5-cup blender container or food processor. Cover. Blend or process until mixture is smooth and creamy, stopping occasionally to scrape down sides of container. Spoon into serving container. Cover. Refrigerate to blend flavors.
*To hard-cook: place eggs in single layer in saucepan. Add enough tap water to come at least 1 inch above eggs. Cover. Quickly bring just to boiling. Turn off heat. If necessary, remove pan from burner to prevent further boiling. Let eggs stand, covered, in the hot water, 15 minutes for Large eggs (about 18 minutes for Extra Large eggs and about 12 minutes for Medium). Immediately run cold water over eggs or place them in ice water until completely cooled.
To remove shell: crackle it by tapping gently all over. Roll egg between hands
to loosen shell. Then peel, starting at large end. Hold egg under running cold water or
dip in bowl of water to help ease off shell.