Preheat your oven to 350°F (175°C). Grease a 9x13-inch baking pan with butter or nonstick cooking spray and set it aside.
Make the caramelized bananas by melting 1/4 cup butter in a large skillet over medium heat. Add the brown sugar and cinnamon, stirring until the sugar dissolves into a smooth syrup, about 2 minutes. Slice the bananas into 1/2-inch thick rounds and add them to the skillet. Cook for 3-4 minutes, gently stirring, until the bananas are soft and coated in a glossy, caramel-colored sauce. Remove from heat and let cool slightly.
In a medium bowl, whisk together the flour, baking soda, and salt until well combined.
In a large mixing bowl, beat the 3/4 cup softened butter and granulated sugar together with an electric mixer on medium speed for 2-3 minutes, until the mixture is light and fluffy. Beat in the eggs one at a time, followed by the vanilla extract and sour cream, mixing just until combined after each addition.
Gradually add the dry flour mixture to the wet ingredients, mixing on low speed until the batter is just smooth with no visible streaks of flour. Gently fold in the slightly cooled caramelized bananas and any syrup from the skillet until evenly distributed.
Pour the batter into the prepared baking pan and spread it into an even layer. Bake for 35-40 minutes, or until a toothpick inserted into the center comes out clean and the top is golden brown. Let the cake cool in the pan on a wire rack for at least 30 minutes before serving.