Preheat the oven to 350°F (180°C). Grease and flour a 9x13-inch (23x33 cm) pan or two 8x8-inch (20x20 cm) pans.
Brown the Butter: In a saucepan, melt 1 ½ cups (340g) of butter over medium heat. Stir as it froths, becoming golden with brown specks at the bottom. Remove from heat and cool slightly. Measure ¾ cup (145g) of the browned butter for the cake, as browned butter loses about 10% of its volume.
¾ cup butter
Prepare Buttermilk Substitute: Combine milk and lemon juice in a measuring cup. Let it sit for 5 minutes to curdle (or use 1 ½ cups/345ml of buttermilk).
1 ¼ cups milk, 2 tbsp lemon juice
Mix Dry Ingredients: In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg. Set aside.
3 cups all-purpose flour, 1 ½ tsp baking soda, ½ tsp ground cinnamon, ¼ tsp salt, ¼ tsp ground nutmeg
Combine Wet Ingredients: Mash the bananas with a fork in a large mixing bowl, then add the brown sugar, white sugar, and eggs. Mix until smooth.
½ cup brown sugar, 1 cup white sugar, 1 ¾ cups bananas, 3 large eggs
Incorporate Butter and Milk: Stir the vanilla into the cooled brown butter. Drizzle it into the banana mixture while mixing. Then, drizzle in the curdled milk and mix until well combined.
1 tbsp vanilla extract
Combine Wet and Dry Ingredients: Gradually add the dry ingredients to the wet mixture, stirring until smooth.
Bake the Cake: Pour the batter into the prepared pan(s). Place in the oven, then immediately lower the temperature to 300°F (149°C). Bake for 60-70 minutes (for a 9x13-inch pan) or until a toothpick comes out clean. Let the cake cool completely before frosting.