How to Make It
Make the cake: Preheat oven to 325°F. Generously grease a 6- to 8-cup Kugelhopf or Bundt pan with butter, and dust lightly with all-purpose flour.
Whisk together almond flour, all-purpose flour, and salt in a medium bowl. Set aside.
Place butter in the bowl of a stand mixer fitted with the paddle attachment. Beat on medium-high speed until pale and creamy, about 3 minutes. Gradually add 1 cup granulated sugar, beating until light and fluffy, about 4 minutes. Scrape down sides of bowl, and add eggs, 1 at a time, beating until well combined after each addition. (Batter may look slightly broken; do not overmix.)
Reduce speed to medium-low, and add flour mixture to butter mixture, one-third at a time, beating for 15 seconds after each addition, stopping as necessary to scrape down sides of bowl to ensure all flour is incorporated.
Scoop 1/2 cup batter into a small bowl, and whisk in 2 tablespoons rum. Add rum batter back to remaining batter, and stir to combine. (Dough may not be completely smooth; do not overmix.)
Spoon batter into prepared pan, and tap pan against counter to distribute evenly. Bake in preheated oven until golden brown and a toothpick inserted in center comes out clean, 42 to 45 minutes.
Make the rum simple syrup: While cake bakes, combine 1/4 cup sugar and 1/4 cup water in a small saucepan. Bring to a boil over medium, and cook, stirring occasionally, until sugar is dissolved. Remove from heat, and stir in 1 tablespoon rum. Return to low heat, and keep warm.
Place cake in pan on a wire rack set inside a baking sheet. Pour 3 tablespoons rum simple syrup over cake. Cool cake 10 minutes. Invert cake onto rack, and brush with remaining syrup. Let cool completely, about 1 hour and 30 minutes.
Make the vanilla icing: Whisk together milk and vanilla bean paste in a small bowl. Place powdered sugar in a medium bowl, and whisk in milk mixture until smooth.
Pour vanilla icing over cake, and let stand until icing is set, about 30 minutes. Serve immediately, or store in an airtight container at room temperature up to 3 days.