Preheat the oven to 350°F (175°C). Line a 9×13-inch pan with parchment paper, leaving overhang on the sides.
Mash the bananas in a bowl until smooth. Add the lemon juice and warm milk. Stir to combine and set aside.
Whisk the flour, baking soda, cinnamon, and salt together in a separate bowl.
Beat the butter, white sugar, and brown sugar on medium-high for 3–4 minutes, until pale and fluffy.
Add the eggs one at a time on low speed, scraping the bowl between each addition. Add the vanilla with the last egg.
Alternately add the flour mixture in 3 additions and the banana mixture in 2 additions, starting and ending with the flour. Mix on low and stop as soon as the flour disappears. Finish folding with a spatula.
Pour the batter into the prepared pan and smooth the top. Bake for 55–60 minutes, until a toothpick inserted into the center comes out clean.
Cool completely in the pan for at least 1 hour.
Beat the cream cheese and butter on medium speed for 2 minutes. Add the vanilla. Gradually add the powdered sugar on low speed, then beat on medium for 1 minute until fluffy.
Spread the frosting over the cooled cake, swirling with a spatula. Slice and serve.