For the cookies:
For the icing:
Grind nuts in food processor until very fine. My little food processor takes a bit under a minute to grind to the right consistency.
Beat eggs and sugar on medium speed until well blended and the mixture is light. Mix spices, flour, baking powder and ground nuts together, and add to egg mixture. Mix on a lower speed until blended. Line a small jelly roll pan (12x16 inches) with parchment paper, pour mixture into pan and smooth.
Bake at 325 degrees for 35 minutes. Remove from oven and let cool for about 10 minutes. Flip pan, remove pan, remove parchment paper from back of Lebkuchen, and flip Lebkuchen upright.
In a small bowl, whisk egg white until foamy (not yet the soft peak stage). Whisk powdered sugar two tablespoons at a time into the egg white. Whisk in vanilla or almond extract. Icing will be fairly stiff but spreadable. Spread on cooling Lebkuchen and allow to cool for another 20 minutes.
Cut into bars about 3 inches by 1 1/2 inches. Store in the ubiquitous airtight container.
Notes: You can omit the nuts and add a bit more flour. Or you may wish to add some (about a half cup?) finely chopped candied fruit. The fruit should probably not be as finely chopped as the nuts.