Place the oven rack in the center position of the oven and preheat the oven to 375°F.
In a large mixing bowl, add the brown sugar, egg, oil, sour cream, milk, and vanilla. Whisk until the mixture is smooth and well blended.
Add the flour, baking powder, cinnamon, and salt to the bowl. Whisk gently until just combined, making sure most of the lumps are mixed out.
Prepare two mini muffin pans by spraying 36 wells with non-stick cooking spray. Scoop 1 ½ Tablespoons of batter into each muffin cup.
Place both pans together on the center rack of the oven. Bake for 10 to 12 minutes, or until the donuts have risen and toothpick inserted comes out clean. (These don't take long to bake, don't over bake.)
While the donuts are baking, melt the butter in a small bowl and set it aside.