To toast the 2 cups walnuts, spread them in an even layer on a baking sheet and toast in a 350°F oven for about 10 minutes, stirring while cooking to ensure even toasting. Allow them to fully cool.
In a food processor, pulse 2 cups all purpose flour, only half of the walnuts, and the 3/4 teaspoon kosher salt until the walnuts are in small chunks, about 4-5 pulses. Transfer this mixture to a separate bowl and set aside.
Run the remaining toasted walnuts through the food processor until finely ground, about 10-20 seconds. Add the 3 tablespoons spiced rum and process until a paste forms which takes only about 5 seconds.
Using a stand mixer with paddle attachment, cream 1 cup unsalted butter and 1/2 cup 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 the dough to plastic wrap, wrap tightly, and refrigerate until firm, at least 1 hour.
Preheat oven to 375°F. Prepare two baking sheets by lining with parchment paper.
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 1 1/2 cups powdered sugar, shaking off any excess.