Preheat the oven to 325 degrees. Lightly grease a donut pan.
In a large bowl, whisk together the flour, sugar, baking powder, nutmeg, cinnamon, and salt.
In a small bowl, stir together the butter milk, egg, and butter.
Add the wet ingredients to the dry and stir with a wooden spoon until fully incorporated.
Fill the donut pan and bake for 8 minutes or until lightly browned and cooked through.
Remove from the oven and cool in the pan for a few minutes before transferring to a wire rack.
To make the glaze:
While the donuts are cooling in the pan, melt the butter in a small saucepan over low heat until it browned, about 8 minutes.
Pour the butter into a small bowl, leaving behind most of the blackened sediment on the bottom of the pan, and whisk in the powdered sugar, milk, and vanilla until smooth.
Dip the donut tops (slightly warm or cool) in the glaze and enjoy.