In a saucepan heat brown sugar, honey, milk, water and cubed butter on medium heat. Stir to combine until the butter is melted and the sugar dissolves, taking it off the heat as soon as it boils.
Add flour, spices, and salt, stirring with a wooden spoon or spatula until combined. Return saucepan to the stove. Continue to heat the mixture and stir while it thickens and starts to pull away from the sides of the saucepan.
Set saucepan aside in a cool place (outside if you live in a cold climate) until the dough is no longer hot so it won't cook the egg white in the next step. But don't chill so long that the dough gets too stiff to stir. Beat the egg white until stiff. When the dough as cooled, stir/knead the stiff egg white into the dough as well as the baking soda and liquor.
Prepare several sheets of parchment paper. Divide the dough into 4 portions. Place each dough portion on a piece of parchment. Place another parchment paper sheet on top and roll the dough out to a disc of about ¼ inch thickness.
Place the rolled out Lebkuchen dough discs in the freezer or fridge until thoroughly chilled through.
Preheat your oven to 350° F and prepare 2 baking sheets with more parchment paper.
Flour your work surface. Then take one chilled dough sheet at a time and, adding only as much flour as needed to keep it from sticking, roll to about ⅛-inch thickness.
Cut out your stars with a star cookie cutter and place on the cookie sheet. Bake one sheet at a time for about 15 minutes. The cookies are done when the tops are opaque and the bottoms slightly browned—be sure to not over bake.
In a small bowl whisk together the egg yolks and water. Pull the finished cookies from the oven, and immediately brush each cookie with a thin layer of the egg wash.