Heat oven to 350° F. Butter and flour three 9 inch round cake pans; set aside.
Heat butter in a saucepan over medium-high heat; cook until it browns. Chill until solid.
In a bowl, whisk together both flours, baking powder and soda, cinnamon, ginger, salt, and allspice. Finely grind coconut and almonds in a food processor. Combine nut mixture, carrots, and flour mixture; mix everything together.
In a bowl, beat together the browned butter and brown sugar until fluffy, 1-2 minutes; beat in yolks one at a time. Add yogurt, juice, zest, and vanilla; beat. Add flour mixture; beat. Wisk egg whites to stiff peaks; fold into batter.
Divide batter between pans; smooth tops with a spatula.
Bake for 30-35 minutes. Let cool slightly.
Glaze & Icing
In a 4 quart sauce pan, combine 4 tablespoons butter, sugar, buttermilk, syrup, scraped vanilla bean pod, and baking soda; boil. Cook until dark (about 3-4 minutes). Remove from heat; remove the vanilla bean pod. Add ½ teaspoon vanilla extract.
Spread glaze over each warm cake. Remove cakes from pans; transfer, glazed side up, to racks. Let cool.
In a bowl, beat remaining butter, mascarpone, vanilla seeds, and cream cheese with mixer until smooth. Add remaining vanilla extract, rum, zest, and sugar; beat. Chill.
Spread 1 ¼ cups icing evenly over 2 cake layers; stack, top with remaining layer.
Frost top and sides; press the chopped walnuts onto the sides. Chill before serving.
