Heat the Oven: Heat your oven to 350℉ (175℃) and position a rack in the middle.
Prepare Rum-Butter Mixture: In a small bowl, combine the melted butter and 3 tablespoons of rum with 1 tablespoon of confectioner's sugar and ¼ teaspoon of salt. Lightly brush the parchment paper on your baking sheets with this rum-butter mixture.Note: The butter/rum mixture should not be incorporated into the cookie dough; it is for brushing the cookies only. Mix the Dry Ingredients: In a medium bowl, combine the almond flour, remaining confectioner's sugar, salt, and ground cinnamon. Set aside.
Mix the Wet Ingredients: In another medium bowl, whisk the egg whites, whole egg, almond extract, vanilla extract, and remaining 2 teaspoons of rum until fully combined and no streaks remain.
Combine the Wet and Dry: Create a small well in the center of the dry ingredients and pour in the egg mixture. Stir with a spatula or wooden spoon until the dough is fully combined and no dry patches remain.
Roll the Cookies: Place the toasted sliced almonds in a shallow bowl. Working in batches, scoop 1½-tablespoon (22-gram) portions of dough and roll them in the sliced almonds, pressing lightly to ensure each ball is fully coated.Tip: You can store the dough in an airtight container for up to 2 days before baking. Transfer the coated dough balls onto the prepared baking sheets, spacing them 1.5 to 2 inches apart. These cookies do not spread much during baking, so shape them as you wish. Brush the tops of the cookies with the rum-butter mixture. Bake and Brush Again: Bake the cookies for 15-17 minutes, or until the edges turn slightly golden and the almonds begin to toast. As soon as you remove them from the oven, brush with the remaining rum-butter mixture.
Cool the Cookies: Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely. Store in an airtight container at room temperature for up to 4 days.