Preheat oven to 350 degrees F. Grease a donut pan with oil and set aside.
In a large mixing bowl, beat the egg and sugar together for about 30 seconds. Add in the vanilla, and buttermilk, oil, and Greek yogurt. Mix together until fully combined.
In a separate bowl, mix the flour, baking soda, baking powder, salt, cinnamon, allspice, and nutmeg together. Add the dry ingredients into the wet, and stir until just combined.
Place about 1/3 to 1/2 of the batter into a piping bag or zip lock bag with one corner cut open. Carefully pipe the batter into the donut pan, making about 6-8 donuts depending on the size of your donut pan. Bake 10-13 minutes or until the donuts are just cooked through.
To make the vanilla glaze, add the powdered sugar, vanilla extract and 2 Tablespoons of milk into a medium bowl. Whisk until smooth. You can add an additional tablespoon of milk if you feel it is too thick.
Let the donuts cool slightly before removing from the donut pan. Dip each donut into the vanilla glaze. Enjoy immediately!