In the bowl of a stand mixer, or using electric beaters, cream together butter, sugar, and brown sugar until fluffy.
Add eggs and vanilla, beating in between each addition. Mix in bananas, buttermilk, and milk.
Add flour, baking powder, baking soda, salt, nutmeg, and cinnamon and mix until just combined, scraping down sides of bowl as needed, until begins to form dough ball. Cover bowl with plastic wrap and let rest in refrigerator for 1-2 hours.
Meanwhile, make cinnamon sugar mixture by combining sugar and cinnamon in shallow bowl.
After resting, heat about 3-4 cups vegetable oil over medium heat in cast iron skillet or stockpot until oil is about 350° F. Use a candy thermometer as you will need to keep your oil at this temp for even frying of the donuts.
Turn out dough onto floured surface and roll out until about 1 1/2 inches thick. Cut out donut shapes with either a donut cutter or large glass and shot glass (for center holes). Gather scraps and roll out again, continuing until all scraps are used up.
Fry donuts in oil for about 2-3 minutes per side. I have found that the easiest way to flip them is to use some chopsticks.
Remove donuts onto paper towel lined plate. While the donuts are still warm and slightly oily, dip each side in the cinnamon sugar mixture to coat evenly.
If you end up waiting until the donut are cool to coat them in cinnamon sugar, then just melt a cup of butter and dip donuts in butter before coating with cinnamon sugar.