Riker and Troi watch on, as Picard chooses a winner.
Captain Picard Day was an annual event held aboard the USS Enterprise-D for the ship's school children. Activities included a contest, judged by Captain Jean-Luc Picard himself, in which the children created paintings and models of the captain.
In 2370, Paul Menegay won the contest with a clay sculpture of the captain's head. Much to Picard's dismay, Commander Riker took great pleasure in the proceedings. As revenge, Picard arranged to have a "Commander Riker Day" the following month and planned to enter the competition himself. (TNG: "The Pegasus")