For the bars: Preheat the oven to 350 degrees F. Cut a piece of foil 9 by 20-inches. Butter the corners and sides of a 9-by-13-inch baking pan. Line the pan with the foil so that the excess hangs over at each of the short sides of the pan.
Whisk the flour with the baking soda and salt in a medium bowl.
Vigorously whisk the bananas with the browned butter in a large bowl, until very well combined and creamy. Whisk the eggs, sugars and vanilla in a medium bowl until smooth. Whisk the egg mixture into the banana mixture.
Fold the flour mixture into the banana mixture with a rubber spatula until well combined. Pour the batter into the prepared pan. Bake until a toothpick inserted in the center comes out clean, about 25 minutes. Cool completely on a rack, about 1 hour.
For the frosting: Beat the butter, cream cheese, vanilla extract and vanilla bean seeds, if using, in the bowl of an electric mixer fitted with the paddle attachment until smooth. Add the confectioners'sugar and beat until fluffy.
When the bar has cooled completely, place a lightweight cutting board or a baking rack over the top of the pan. Flip and set on the counter. Gently pull the pan from the foil and the foil from the pan to remove. Gently peel the foil off the bar. Place another lightweight cutting board on top, and flip the bars right-side-up.
Frost the bars all the way to the edge and refrigerate 20 minutes before slicing into 1 1/2-by-3-inch bars.
Cook's Note: We like these bars even better cold the following day. To store in the refrigerator, chill the frosted bars about 30 minutes, uncovered, and when the frosting is set wrap in plastic wrap.