Preheat the oven to 425˚F (218˚C). Lightly grease a mini donut pan (If you have 2 pans, grease 2 pans.).
In a small bowl, mix the sugar and lemon zests together. Add brown sugar. Set aside.
In a large mixing bowl with an electric mixer fitted with a paddle attachment, beat the butter for 2 minutes. Add lemon sugars and vegetable oil. Beat until fluffy, about 3 minutes. Add egg and beat to combine. With a spatula, stir in the baking powder, baking soda, salt and vanilla. Mix well.
Continue using a spatula, stir and add the flour and milk to the butter mixture alternately in 3 additions, by starting and finishing with the flour. Mix until just combined. Do not overmix.
Spoon the batter into a pipping bag or Ziploc bag, fill the donut pan with batter all the way up to the rim.
Bake in the oven for about 6 – 8 minutes. Test the donuts with a toothpick for doneness. The toothpick should come out clean. Remove from oven and cool in the pan for 5 minutes. Transfer the donuts to a cooling rack and cool completely. If you have extra batter, clean and dry the donut pan. Lightly grease and repeat filling the pan.
For the glaze, in a medium bowl, add powdered sugar. Add the milk, 1½ tablespoons lemon juice & pink food coloring (if using). Whisk until smooth. Add more lemon juice if the frosting is too thick. Glaze the donuts and let them set for 20 minutes.