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In the bowl of a food processor fitted with a steel S blade, grind the oats to a flour. Add the brown sugar, flour, salt, baking soda, baking powder, cinnamon, and nutmeg and process briefly to blend. Toss in the butter cubes and process until the butter is well incorporated and the dough begins to clump and pull off the sides of the bowl. Add the egg yolk and pulse until the dough comes together with no dry pockets.
Line a work surface with a large sheet of plastic wrap. Turn out the dough onto the plastic wrap and form it into a log about 10-inches/25cm long and 1 1/2 inches/3.8 in diameter. Wrap the dough tightly. Chill the dough until firm, about 3 hours in the refrigerator.
Position a rack to the center of the oven and preheat it to 350 degrees F. Line 2 baking sheets with parchment paper. Use a thin, sharp knife to slice the dough log into 2 dozen rounds, each just shy of 1/2-inch/1.25 cm thick. Transfer the rounds, evenly spaced about 2 inches/5cm apart, to the prepared baking sheets. Bake until golden and firm on the edges with a bit of give in the centers, about 12 minutes. Transfer the cookies to a wire rack to cool completely—they will crisp all the way through as they cool.
In a small bowl, combine the egg white, powdered sugar, vanilla and salt. Mix with a handheld electric mixer until smooth and thick. (Alternatively, you could also do this in a stand-up mixer or in a medium bowl by hand.) Blend in 1 teaspoon of water until smooth. To ice the cookies, working 1 at a time, kiss the tops lightly to the surface of the icing—you want to just show the cookie to the icing, not submerge it.
Let the excess icing drip off for a moment, and then set the iced cookie on a wire rack. If the icing begins to firm while you’re dipping, loosen it with a few drops of water. Allow the icing to dry completely before serving, about 1 hour.
Store in an airtight container for up to 1 week.