Step 1: Preheat the Oven - Start by preheating your oven to 350°F (175°C). This ensures that your donuts bake evenly.
Step 2: Mix Wet Ingredients - In a large mixing bowl, whisk together the egg, light brown sugar, granulated sugar, canola oil, sour cream, vanilla extract, and mashed ripe bananas until smooth and well combined.
Step 3: Combine Dry Ingredients - In another bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Gradually add the dry ingredients to the wet mixture, stirring until just combined. Be careful not to overmix!
Step 4: Fill the Donut Pan - Spoon the batter into a piping bag or a zip-top bag with the corner cut off. Pipe the batter into the prepared donut pan, filling each cavity about ¾ full.
Step 5: Bake the Donuts - Bake for 10-12 minutes, or until a toothpick inserted into the center comes out clean. Allow the donuts to cool in the pan for a few minutes before transferring them to a cooling rack.
Step 6: Prepare the Browned Butter Caramel Glaze - In a small saucepan over medium heat, melt the unsalted butter until it starts to foam and turn golden brown, about 5 minutes. Remove from heat and stir in the light brown sugar, half-and-half, confectioners' sugar, vanilla extract, and salt. Whisk until smooth, adjusting the consistency with additional half-and-half if necessary.
Step 7: Glaze the Donuts - Once the donuts are completely cool, dip the tops into the glaze or drizzle it over them. Allow the glaze to set for a few minutes before serving.