Prep. Preheat the oven to 350˚F. Grease a doughnut pan with cooking spray.
Combine the dry ingredients. In a large bowl whisk together the flour, sugar, baking powder, and salt; set aside.
Warm the milk and melt the butter. In a small saucepan, combine the milk, rum, egg, and butter. Cook over medium heat, stirring constantly with a whisk until the butter melts, about one to two minutes.
Add the liquid to the dry ingredients. Remove from the heat and add the milk mixture to the flour mixture; mix until combined.
Transfer the batter to the doughnut pan. Using a large tablespoon, spoon the batter mixture into the pan.
Bake. Pop the pan in the preheated oven and bake for 12 to 14 minutes, or until a toothpick inserted in the center of a donut comes out clean.
Cool. Remove the donuts from the oven and immediately invert the pan onto a cooling rack.
Make the glaze. While the doughnuts are cooling, begin to work on the glaze. Place the milk, lime juice, and powdered sugar in a small saucepan and cook over medium heat, stirring constantly until combined and no lumps appear.
Dip the doughnuts. Remove the glaze from the heat and set the saucepan over a bowl of hot water to keep it warm. Stir in the dried mint. Dip each doughnut into the glaze, one at a time, and place it glaze-side up on a wire rack. Set a plate or wax paper underneath the rack to catch any drips.