In a large bowl, mix together the Greek yogurt, milk, sugar, egg, vegetable oil, and vanilla.
Spoon the mixture into the donut pans until each donut cup is 2/3 of the way full of batter.
Bake for 8 to 10 minutes, then remove from oven and allow to cool in pan for 5 minutes. Remove from pan and cool completely on a wire cooling rack.