In a large saucepan bring coconut milk and ginger to a simmer. Remove from heat; add basil. Cover; let stand 1 hour. Strain milk, discarding solids. Chill at least 2 hours or overnight.
In a blender combine avocados and 1 cup of the strained coconut milk. Cover; blend until very smooth. Transfer to a large bowl. Whisk in remaining strained coconut milk, lime juice, and salt. Top with additional basil, chopped avocado, and chili oil or olive oil.