Preheat the oven to 350°F. Lightly grease a doughnut pan and set aside.
Doughnuts: In a medium bowl, mix together the flour, baking powder, baking soda, salt, cinnamon, nutmeg and ginger. Set aside.
In a medium saucepan, melt all of the butter (4 tablespoons in total). Keep watching the butter until it starts to get brown and give off a nutty smell. Remove from heat immediately.
Meanwhile, in a small bowl, mix together the milk and vinegar. Set aside to “sour”.
Beat together the “soured” milk, egg and vanilla. Measure out 2 tablespoons of brown butter and mix it into the wet ingredients.
Slowly mix in the wet ingredients to the dry ingredients until just combined. Fold in the grated apples. Do not overmix!
Gently scoop in the batter to the prepared doughnut pan, and bake for 8 to 10 minutes, or until golden brown.
Glaze: While the doughnuts cool, mix together the powdered sugar and salt. Mix in the remaining brown butter, vanilla and milk until fully combined.
Once the doughnuts are completely cool, spread or dip the doughnuts in the icing. Place in the fridge for at least 15 minutes for the icing to harden.