Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line with parchment paper.
Mash the bananas in a bowl until smooth. Set aside.
Whisk the dry ingredients: In a medium bowl, combine 1 ¼ cups flour, baking soda, salt, and cinnamon (if using). Set aside.
Cream the butter and sugar: In a large bowl, beat the softened butter and granulated sugar together with an electric mixer until light and fluffy, about 2-3 minutes.
Add the egg and vanilla: Beat in the egg and vanilla extract until fully incorporated.
Add the mashed bananas: Stir in the mashed bananas until evenly combined.
Mix the wet and dry ingredients: Add the dry ingredients to the wet mixture in two batches, alternating with the buttermilk. Start and finish with the flour mixture. Stir until just combined. If using, fold in the sour cream or yogurt for extra moisture.
Bake the cake: Pour the batter into the prepared cake pan and smooth the top. Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
Make the brown butter frosting: In a small saucepan, melt ⅓ cup butter over medium heat. Once melted, continue cooking, swirling the pan, until the butter turns golden brown and smells nutty (about 4-5 minutes). Remove from heat and allow it to cool for a few minutes.
Frost the cake: In a medium bowl, beat the browned butter with powdered sugar, vanilla extract, and a pinch of salt. Add milk, one tablespoon at a time, until you reach your desired frosting consistency. Once the cake is completely cool, spread the brown butter frosting on top.