Gather the ingredients. Preheat the oven to 350°F.
Peel and cut the peaches into thin slices. Lightly grease your baking dish and arrange half of the peach slices on the bottom.
Combine the brown sugar, cinnamon, and nutmeg. Sprinkle half of the brown sugar mixture over the peaches in the baking dish.
Drizzle rum and melted butter over the peaches and sugar.
In a separate bowl, beat the butter and cream cheese on medium-high speed, scraping down the sides of the bowl as needed. Then, add the sugar and mix until light and fluffy.
Add the eggs one at a time, mixing well after each addition. Beat the mixture for 30 seconds after adding each egg.
Add the milk and vanilla extract to the mixture and stir to combine.
In another bowl, whisk together the dry ingredients until well combined.
Add the flour mixture to the cream cheese mixture, half a cup at a time, mixing on low speed. Scrape down the sides of the bowl halfway through and continue mixing until just combined. Do not overmix.
Spread half of the batter over the peaches.
Arrange the remaining peaches over the batter and sprinkle with the remaining cinnamon-sugar mixture.
Spread the remaining batter over the peach layer. Smooth it out with a spatula and bake for 45 to 50 minutes or until golden brown.
For the icing, whisk together the milk, powdered sugar, and vanilla extract until smooth and creamy.
Allow the peach cobbler pound cake to sit for 10 minutes, then invert it onto a wire rack to finish cooling. Transfer to your serving plate.
Spoon the vanilla icing over the cooled pound cake and garnish with mint leaves.