Using only the hardboiled yolk (discarding the white), press through a fine mesh strainer. In the bowl of a stand mixer, combine strained yolk, butter, sugar and salt. Mix until fluffy and light. Add vanilla and mix again. Add flour and cocoa and mix well.
Gather dough together, divide dough in half and form two logs, about 6" long and up to 2" round or squared off. Wrap in plastic wrap and chill for 45 minutes in freezer or 2 hours or overnight in refrigerator.
Preheat oven to 350°. Cut ¼" cookie slices from log, brush with egg white and water mixture, and sprinkle with sugar. Bake 11 to 13 minutes (they will start to crack on top). Allow to cool completely before removing to wire rack. These cookies are very tender and will fall apart if treated roughly.
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