Preheat the oven to 350ºF and spray a donut pan with nonstick cooking spray.
Combine the dry ingredients in a medium bowl and set aside.
Using an electric mixer or stand mixer, cream the butter, sugar, and vanilla together until the mixture is light in color and fluffy. Add the mashed banana, maple syrup, and Greek yogurt. Mix again.
Add one egg at a time to the bowl and mix until combined.
Slowly add the dry ingredients to the wet ingredients until the batter is a muffin or cake batter consistency.
Transfer the donut batter into a piping bag or large gallon-sized bag. Snip the end of the bag and pipe the donut batter into the donut pan, filling each donut ¾ of the way full.
Bake for 7-8 minutes or until fully baked. Remove from the oven and let the donuts rest for 2-3 minutes. Remove the donuts from the pan and let them cool on a wire rack until cool. Most donut pans have to bake 6 donuts at a time, so you will most likely have to bake another batch, as this recipe makes 12.
While the donuts cool, make the brown butter glaze. Heat a small skillet over medium/high heat. Add the butter and let it melt until it begins to turn golden brown and little brown specs begin to form. Remove from heat immediately and transfer it to a mixing bowl.
Add the bananas, vanilla, and salt to the butter and mix. Add the powdered sugar until a drippy glaze forms. If the mixture is too thick, add 1-2 tablespoons of milk until it reaches a thick but drippy consistency. If the mixture is too thin, add 1-2 tablespoons ore of powdered sugar.
Use a knife or a spoon to spread the glaze over the top of each donut. Eat immediately or store in an airtight container in the refrigerator.