Preheat oven to 350℉.
Place rum butter ingredients (melted butter, rum, powdered sugar, and salt) in small bowl and combine. (You will use the rum butter three times: to brush the parchment paper, to brush the cookies before you bake them, and to brush the cookies after you bake them.)
Place dry ingredients (almond flour, powdered sugar, salt) in large bowl and whisk together until no lumps remain.
Place wet ingredients (egg whites, egg, rum, and almond extract) in small bowl and beat until no streaks remain.
Make well in center of dry ingredients. Pour in egg mixture. Mix with a spatula or wooden spoon until no dry spots remain. (Dough can be stored in fridge for up to 2 days before baking.)
Line 2 baking sheets with parchment and brush lightly with rum butter.
Pour sliced almonds into shallow bowl.
Scoop out dough, 1 tablespoonful at a time. Shape tablespoon of dough into ball, roll in almonds, and press slightly so dough is thoroughly coated with almonds.
Place cookies on baking sheet, two inches apart.
Brush tops on cookies with more rum butter.
Bake cookies until slightly cookies turn golden at the edges and almonds start to toast, about 20-22 minutes.
Immediately after removing cookies from oven, brush with more rum butter and top with flaky sea salt.
Let cookies cool 5 minutes before transferring from baking sheet to wire rack. Cool completely.
