To toast the walnuts, spread an even layer on a baking sheet and toast in a 350 degree F oven for 10 minutes, stirring them while cooking to ensure even toasting. Allow to fully cool.
In a food processor, pulse flour, only half of the walnuts, and the salt until the walnuts are in small chunks, 4-5 pulses. Transfer to a separate bowl and set aside.
Run the remaining toasted walnuts through the food processor until finely ground 10-20 seconds. Add the spiced rum and process until a past forms which takes only about 5 seconds.
Using stand mixer with paddle attachment, cream butter and granulated sugar until light and fluffy, about 5 minutes. Add the walnut rum paste and beat until combined. Turn the speed to low, add the flour mixture one spoonful at a time, and stop mixing as soon as everything has combined. Do not over mix.
Transfer dough to plastic wrap, wrap tightly, and refrigerate until firm, at least 1 hour.
Preheat oven to 375 degrees F. Prepare two baking sheets with parchment paper or a silpat.
Form dough into walnut sized balls. Bake until bottoms just start to brown, about 12-14 minutes. Allow to cool on sheets for about 10 minutes, then transfer to a cooling rack. Repeat until all the cookies have baked.
Prior to eating, roll fully cooled cookies in powdered sugar, shaking off any excess.