Place eggs in a large pot with cold water, covering eggs by 1 inch. Bring to a full boil; boil 1 minute. Cover pot and remove from heat. Let sit, covered, for 15 to 17 minutes. Drain and run cold water over the eggs. Crack eggs and let sit in cold water 10 minutes or until cool. Drain, then peel eggs.
Cut eggs in half lengthwise, then carefully remove the yolks, leaving the whites intact. Place yolks in a bowl; mash with 1/3 cup mayonnaise, avocado, mustard, shallot, lemon juice, and salt until blended. Using 2 teaspoons or a small ice-cream scoop, scoop yolk mixture into small balls and position in hollows of whites to resemble eyeballs.
Press an olive slice into center of each yolk eyeball. Stir together 3 tablespoons remaining mayonnaise and a few drops of red food coloring in a small bowl for the "blood."
Transfer colored mayonnaise mixture into a small seal-able bag and snip the corner to form a pastry bag. Decorate eyeballs with bloodshot veins. Cover and refrigerate up to 6 hours before serving.
Make Ahead: Egg yolk mixture can be prepared 1 day ahead and refrigerated. Fill egg whites, decorate and refrigerate up to 6 hours before serving.
