Place rack in the centre of the oven and preheat to 350F. Grease a standard size, 12 cavity donut pan with cooking spray and set aside.
In a large bowl, whisk together the flour, brown sugar, 1/2 tsp cinnamon, baking powder, baking soda and salt.
In a separate medium bowl, stir together the yogurt, milk, oil, egg and vanilla until smooth. Pour the wet ingredients over the dry ingredients and mix with a rubber spatula until just combined and no streaks of flour remain. Be careful not to over mix to ensure tender results. The batter will be very thick.
Spoon the batter into the donut pan, filling each cavity 3/4 of the way full. (You can also spoon the batter into a large resealable bag and snip a corner of the bag to make a pastry bag. Squeeze the bag to pipe batter into each donut pan cavity.)
Bake for 9-10 minutes, until batter is set and edges are lightly browned. Allow the donuts to cool in the pan for 5 minutes while you prepare the cinnamon sugar coating.
To make the cinnamon sugar coating: Place a cooling rack over top of a baking sheet and set aside (the baking sheet will catch the excess cinnamon sugar, making clean up easier). In a small bowl, combine 1/2 cup of sugar and 1/2 tsp of cinnamon. In a separate small bowl, melt the butter in the microwave. Working with one donut at a time, dip the donut into the butter and roll it around to coat evenly. Immediately dunk the donut into bowl of cinnamon sugar and turn it to coat the donut all over. Transfer the donut to the prepared cooling rack. Repeat with the other donuts until all donuts are coated in the cinnamon sugar. Serve immediately.