Place butter in the bowl of a food processor (11-cup or larger suggested). Pulse a few times until creamy. Add the sugars and pulse until combined. Next add the egg, egg yolk, rum, vanilla, and 1/2 teaspoon of the nutmeg. Pulse until completely combined and creamy, but not fluffy. This will help keep create the perfect sandy texture when the cookies are baked.
To the food processor, add the Pamela's All-Purpose Flour and sea salt. Pulse a few times, then process for 30-60 seconds or until well-combined. The dough will be fairly sticky. If needed, scrape down the sides and pulse again a few times.
Turn the dough out onto a clean surface lined with plastic wrap, parchment or waxed paper; divide evenly into two portions. Form each half into a log that is about 2" in diameter and about 12" in length. (It doesn't have to be very precise.) Wrap well and refrigerate until firm, about 2 hours.
To bake, adjust oven racks to be able to bake two sheets at a time. Preheat oven to 350°F. Line two rimmed baking sheets with parchment paper. Set aside.
Working in batches, unwrap the dough. Mix together coarse or raw sugar and remaining 1/2 teaspoon nutmeg. Brush with a little of the lightly beaten egg white over the top and sides of the dough. Sprinkle with the nutmeg sugar. Turn dough over and brush remaining side and sprinkle with more of the nutmeg sugar. Gently press the sugar into the dough if it's not sticking very well.
Using a sharp, thin paring knife, cut the dough into rounds that are about 1/4" thick. Arrange on the prepared baking sheets leaving a little room between the cookies. They will not spread very much. Repeat with remaining dough.
Place one baking sheet on each of the oven racks. Bake for 8 minutes, then switch the pans from top to bottom. This will ensure even baking. Bake for another 7-8 minutes or until cookies are lightly golden and look set in the middle. It's better to err on the side of baking them longer or they won't be crisp.
Allow cookies to cool completely on baking sheets. Remove and place in an airtight container. Will keep for several weeks.
-100% palm shortening or non-hydrogenated shortening (blend of several oils) can be substituted. The dough will be slightly firmer. If it's not sticking together or feels too dry, add a little bit of cool water until it's soft and pliable. (I don't recommend other brands of shortening, but they will certainly work too.)
-Vegan margarine (like Earth Balance Buttery Sticks, Melt, or others) can be used. Due to the higher water content, the dough may be a little more sticky. If too sticky, add more Pamela's All-Purpose Flour a tablespoon at a time until it reaches the right consistency.
-Other sweeteners can be substituted, but the texture and color of the baked cookies will differ. If using coconut sugar, place it into a blender to finely grind it so it mixes more easily into the butter.
-I'm wary to suggest using chia or flax eggs here because the cookies will come out gummy.
-1/2 teaspoon rum or brandy extract plus enough water to make 2 tablespoons can be substituted for the rum.
-Dough can be prepared and frozen, wrapped well in parchment, then aluminum foil or placed in a freezer-safe container for several months.