For the Cake

1. Position an oven rack in the center of the oven and preheat to 325°F. Spray a 12-cup Bundt pan with nonstick cooking spray with flour, such as Baker's Joy or Pam with Flour. (Alternatively, grease the pan with butter and then dust with flour.)
2. In a medium bowl, whisk together the eggs, egg yolks, rum, vanilla extract and almond extract. Set aside.
3. Place the cake flour, brown sugar, granulated sugar, baking powder, and salt in the bowl of an electric mixer fitted with the paddle attachment; mix on low speed for about 30 seconds to combine. Add the butter and beat on low speed for about 2 minutes, or until the ingredients are well combined. (The mixture will look a bit like cookie dough.)
4. Add a third of the liquid ingredients and mix on low speed until just incorporated. Increase the speed to medium and beat for 2 minutes, then stop the mixer and scrape down the sides and bottom of the bowl with a rubber spatula. Add another third of the liquid and mix on low speed until just combined. Increase the speed to medium and beat for two minutes, then stop the mixer and scrape again. Add the remaining liquid and beat and scrape as before.
5. Transfer the batter to the prepared pan and smooth the top. Bake for 65 to 75 minutes, or until a toothpick inserted halfway between the central tube and side of the pan comes out clean.
6. Cool the cake on a rack for ten minutes.
7. While the cake bakes, make the glaze. Melt the butter in a small saucepan. Add the water and sugar and bring to a boil. Turn the heat down to medium-low and simmer, stirring frequently, for about 5 minutes, or until the mixture is slightly thickened. Remove from the heat and stir in the rum and salt.
8. Using a toothpick or thin skewer, poke holes all over the bottom of the cake about ½-inch apart, going about ¾ of the way down (you will feel like you are butchering the cake; don't worry, it will be fine). Brush half of the glaze over the cake and let it soak in. If the glaze pools on the surface, poke more holes to help it sink in. Invert the cake onto a cake platter. Gradually brush the remaining glaze evenly over the top and sides of the cake, letting it soak in as you go. (Go slowly so that the glaze gets absorbed, rather than drips off the sides.) Let cool completely before serving.
9. Make-Ahead/Freezer-Friendly Instructions: The cake can be made up to 1 day ahead of time and stored in a cake dome or covered container at room temperature. It can also be frozen for up to 3 months. After it is completely cooled, double-wrap it securely with aluminum foil or plastic freezer wrap, or place it in a heavy-duty freezer bag. Thaw overnight on the countertop before serving.
10. Note: It's important to use cake flour in this recipe. All-purpose flour will make the cake dry and dense.