Preheat oven to 350.

In a large bowl whisk together the flour, sugar, baking powder, and salt; set aside.

In a small saucepan bring together the milk, rum, egg, and butter.

Cook over medium-heat, stirring constantly with a whisk until the butter melts, about one to two minutes.

Remove from heat and add the milk mixture into the flour mixture; mix until combined.

Grease a doughnut pan with cooking spray.

Using a large tablespoon, spoon the doughnut mixture into the pan.

Bake for 12 to 14 minutes, or until a toothpick inserted comes out clean.

Remove and immediately invert the pan onto a cooling rack.

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 just until combined and no lumps appear.

Remove from heat and set the saucepan over a bowl filled with hot water.

Stir in the dried mint.

Dip the doughnuts into the glaze, one at a time, and set on a rack, glaze side up. Put a baking sheet underneath the rack because some of that glaze will drip off.