Preheat oven to 350°F. Lightly grease a doughnut pan with nonstick cooking spray and set aside.
Melt 3 tablespoons butter in a small saucepan over medium-low heat, stirring occasionally. Butter will crackle and foam a bit as it browns. Once butter is light brown and smells nutty, remove from heat and immediately transfer to a small bowl. Be careful, it goes from brown butter to black burnt butter quickly.
In a medium mixing bowl, whisk together flour, sugar, baking powder, baking soda, salt, and cinnamon.
In a small mixing bowl, whisk together egg, buttermilk, and vanilla extract. Measure out 2 tablespoons of the browned butter and whisk into the wet ingredients.
Add wet ingredients to dry ingredeints. Stir together until ingredients are well combined, but do not overmix.
Spoon batter into a large ziploc bag and use a scissor to cut about a 1/2″ piece off one corner to create a makeshift disposable pastry bag. Fill each well in the doughnut pan approximately 3/4 full. Alternatively, use a spoon to fill donut pan and smooth out batter.
Place in the oven and bake for 9-12 minutes, until a toothpick inserted into doughnut comes out clean. Remove from the oven and allow to cool in the pan before inverting onto a wire rack to cool completely.
Combine sugar, brown, sugar, and cinnamon in a small bowl. Melt remaining 3 tablespoons butter in another small bowl. Working one at a time, lightly brush doughnut with butter, and roll around in sugar mixture until evenly coated.
Devour immediately or store at room temperature in a ziploc bag. But really, devour immediately.