Position knife blade in food processor bowl; add first 6 ingredients. Process until smooth; set aside.
Beat cream cheese and mayonnaise at medium speed of an electric mixer until smooth. Press shredded cucumber and onion between paper towels. Add cucumber and onion to cream cheese mixture. Stir in 1/8 teaspoon salt and 1/8 teaspoon red pepper. Set aside.
Spread butter on one side of each bread slice. Spread 1 tablespoon plus 1 teaspoon avocado mixture on buttered side of each of 10 white bread slices; top each with 1 whole wheat slice, buttered side down. Reserve remaining avocado mixture. Spread about 3 tablespoons cucumber mixture on each stack; top each with 1 white bread slice, buttered side down. Wrap in plastic wrap; chill. Spoon remaining avocado mixture into a decorating bag fitted with metal tip No.
Unwrap sandwiches, and remove crusts, using an electric knife; discard crusts. Cut each sandwich into fourths, using electric knife. Pipe a rosette of avocado mixture onto each sandwich. Garnish with cucumber wedges, if desired.