1. For the cake: Preheat the oven to 375 degrees F. Grease two 9-inch cake pans with butter. Line the pans with parchment paper, and then grease the parchment. Set aside.
2. Whisk together the flour, baking powder and salt in a medium bowl and set aside. Mix the milk, rum and vanilla in a spouted measuring cup and set aside. Place the butter and sugar in the bowl of a stand mixer with the paddle attachment, or using a large bowl with a hand mixer, beat on medium-high speed until very light and fluffy, about 5 minutes. Add the eggs one at a time, mixing well after each egg. Turn the mixer to low, add half of the flour mixture and mix until incorporated. With the mixer still running and set on the lowest setting, slowly stream in the milk mixture. Increase the speed to medium low and mix until well combined. Don't worry if the batter looks a little broken; it will come back together. Add the remaining flour and mix on low until combined. Using a rubber spatula, fold the batter a few times to ensure all of the ingredients are incorporated.
3. Divide the batter between the prepared cake pans. Bake until set and golden around the edges, 25 minutes. At the halfway point, swap the positions of the pans in the oven so that they bake evenly. Cool in the pans for 5 minutes, and then remove to a rack to cool completely.
4. For the sweetened whipped cream: Add the heavy cream, sugar and vanilla to the bowl of a stand mixer with the whip attachment, or a medium bowl using a hand mixer, and whip on high to form soft peaks. Refrigerate for at least 15 minutes.
5. For the glaze: Stir together the sugar, heavy cream, rum and egg yolks in a medium saucepan over medium heat, stirring constantly, until the mixture coats the back of the spoon, about 5 minutes. Stir in the pecans. Continue to cook and stir until it turns golden and thickens, about 5 more minutes. Stir in the vanilla. Remove from the heat and allow to cool.
6. To assemble the cake: Remove the parchment from the layers and place one layer on a plate. Spoon half of the glaze, about 1/2 cup, over the layer and spread to the edges. Place the second layer on top and spoon the remaining glaze over the top, allowing some of the glaze to gently drip down the sides if desired.
7. To serve, put the chilled whipped cream in a piping bag with a large star tip and decoratively pipe around the edges. Alternatively, cut into individual slices and dollop some whipped cream on top of each slice.