Preheat the oven to 375°F. Butter and line a 9-by-13-inch pan with parchment. Butter the parchment.
Beat the butter and 1½ cups of the brown sugar with an electric mixer on medium-high until light and fluffy, 5 minutes. Scrape down the sides of the bowl and add the vanilla and eggs, one at a time. Beat to combine.
Whisk together the flour, baking powder, and salt. Add to the batter, alternating with the sour cream, beginning and ending with the flour mixture. Beat until just combined.
Spoon the batter into the prepared pan and smooth with a spatula. Lay the bananas, cut-side up, across the batter (they will sink in) and sprinkle with the remaining brown sugar. Bake until a toothpick inserted in the center of the cake (avoiding the bananas) comes out clean, 35 to 45 minutes. Serve warm with vanilla ice cream.