Preheat the oven to 350 degrees F (175 degrees C). Butter a 10x15-inch rimmed baking pan.
Beat 1/2 cup butter and white sugar together in a bowl using an electric mixer until creamy. Beat in bananas, eggs, and 1 teaspoon vanilla extract.
Combine flour, baking soda, and salt in a separate bowl. Add to the banana mixture and mix well. Pour into the prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes. Cool on a wire rack for 5 minutes. Run a table knife around the edges to loosen. Invert carefully onto a serving plate or cooling rack. Let cool, about 30 minutes.
Beat cream cheese and 1/2 cup butter together in a bowl using an electric mixer until fluffy. Add confectioners' sugar and 2 teaspoons vanilla extract gradually; beat well. Spread frosting over the cooled bars and sprinkle walnuts on top.