Preheat and prep: Heat your oven to 350°F (175°C). Lightly grease your doughnut pan with cooking spray or a thin brush of oil.
Mix dry ingredients: In a medium bowl, whisk the flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Set aside.
Whisk wet ingredients: In a large bowl, whisk the mashed bananas with the granulated sugar and brown sugar until combined. Add the egg, yogurt, oil, and vanilla. Whisk until smooth and glossy.
Combine batter: Add the dry ingredients to the wet bowl. Stir gently with a spatula until just combined. Do not overmix; a few small lumps are fine. Overmixing can make the doughnuts tough.
Fill the pan: Spoon the batter into the cavities, or transfer it to a piping bag (or zip-top bag with a corner snipped) and pipe it in. Fill each well about 3/4 full so they rise without overflowing.
Bake: Bake for 10–12 minutes, or until the tops spring back when lightly pressed and a toothpick comes out clean or with a few moist crumbs. The edges should be lightly golden.
Cool: Let the doughnuts rest in the pan for 5 minutes, then turn them out onto a cooling rack. Cool completely before glazing to prevent the glaze from sliding off.
Make the glaze: In a small bowl, whisk the powdered sugar, 2 tablespoons milk, vanilla, and a pinch of salt until smooth. If it’s too thick, add milk in tiny splashes. If too thin, whisk in a bit more powdered sugar. You want a pourable, opaque glaze that clings.
Glaze the doughnuts: Dip the top of each cooled doughnut into the glaze, let excess drip off, and place back on the rack. For a thicker shell, let the first coat set for a few minutes, then dip again.
Set and serve: Let the glaze set for 10–15 minutes. Enjoy slightly warm or at room temperature.