Preheat the oven to 375 F. Grease a 9x13-inch pan with nonstick cooking spray; set aside.
In a medium bowl, whisk together the flour, baking soda, and salt; set aside.
In a large mixing bowl, beat together the granulated sugar and softened butter with an electric mixer until light and fluffy. Add the eggs and beat until well combined.
Beat in the sour cream, mashed bananas, and vanilla extract.
Add the dry mixture to the wet and stir just until combined.
Pour the batter into the prepared baking pan and spread out edge to edge.
Bake at 375 F for 25-30 minutes, until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and cool to room temperature.
For the frosting: melt the butter in a skillet over medium heat and continue to cook until the butter begins to turn a light amber brown.
Remove the pan from the heat and whisk in the powdered sugar and vanilla. The frosting will be thick and chunky.
Add 2 T of heavy whipping cream and whisk until the frosting is smooth, adding more cream 1 T at a time until the desired consistency is reached.
Spread the frosting over the completely cooled cake.