Preheat your oven to 350°F and grease a 9x13 inch baking pan. In a large mixing bowl, cream together the butter and sugar until smooth. Slowly add in the rum, mixing well. In a separate bowl, combine the flour, baking powder, and salt. Gradually add the dry ingredients to the wet mixture, alternating with the milk. Mix until well combined.

Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 35-40 minutes, or until a toothpick inserted in the center comes out clean. Allow the cake to cool in the pan for a few minutes before transferring to a wire rack to cool completely.

Once your cake has cooled, it's time to make the rum glaze! In a small saucepan, melt 1/2 cup butter and 1/2 cup brown sugar over medium heat. Once the butter and sugar have melted, stir in 1/2 cup rum and bring the mixture to a boil. Reduce the heat to low and simmer for 2-3 minutes, or until the glaze has thickened slightly.

Pour the rum glaze over the top of the cooled cake, allowing it to drizzle down the sides. Serve immediately, or store in an airtight container for up to 3 days.