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.
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).
Mix Dry Ingredients: In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg. Set aside.
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.
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.
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.