Preheat the oven to 350 degrees. Coat two 24-cup mini muffin pans (or 3 12-cup mini muffin pans) with cooking spray.
To make the donuts, in a large bowl, combine the flour, 1/2 cup sugar, 1 1/2 teaspoons cinnamon, baking powder, baking soda, and salt. Mix well to combine.
In a separate bowl, whisk together the yogurt, oil, egg, and vanilla.
Add the yogurt mixture to the flour mixture and mix with a spatula until just blended (try not to over-mix). The batter will be thick. Spoon the batter into the prepared pans, filling each cup 3/4 full (no higher!).
Bake for 8 to 10 minutes, until a wooden pick comes out clean or with little moist bits clinging to it. Cool the donuts, in the pans, on wire racks.
To make the coating, place the melted butter in small bowl. In a separate bowl, mix together the 3/4 cup of sugar and 2 teaspoons of cinnamon.
Dip each donut hole in the melted butter and turn to coat (coat lightly). For an even lighter coating, brush the butter on with a pastry brush. Transfer the donut holes to the cinnamon-sugar mixture and turn to coat. If you want easy cleanup, transfer the finished donuts to a parchment-lined baking sheet as you work.
Enjoy warm or room temperature. Store leftovers in an airtight container at room temperature (don't refrigerate or the moisture from the fridge will cause the donuts to absorb the sugar coating).