Preheat oven to 350°F and grease a standard 9x13" pan.
Using a mixer, cream the butter and granulated sugar until light, about 2 minutes.
Add eggs, one at a time.
Slowly add in mashed bananas.
Add vanilla to sour milk or buttermilk and stir slightly to combine.
Sift or whisk together the baking powder, baking soda, salt and flour in a separate bowl.
Alternate adding flour mixture and milk to the batter in three additions, starting and ending with the flour mixture. Only mix until just combined, do not overmix.
Pour into greased pan and bake for 30 minutes, or until lightly golden brown and a toothpick inserted in the middle of the cake comes out clean.
Allow pan to cool on a wire rack at least 30 minutes before frosting.
For the Frosting
In small sauce pan, combine all ingredients except for the vanilla and powdered sugar.
Over medium heat, bring slowly to a full boil, whisking constantly.
When the mixture reaches a full boil, boil briskly 1 minute, whisking constantly.
Remove from heat and whisk in vanilla.
Allow to sit until cools slightly, about 20 minutes.
Then, whisk in powdered sugar and beat until smooth. Frost banana cake immediately before frosting sets.
Notes
To make your own sour milk for this recipe: add 2 teaspoons of white vinegar to milk. You can also substitute buttermilk.
Once the frosting comes to a full boil, stir it constantly to prevent it from burning
If the frosting becomes too thick, you can easily thin it out with a splash of heavy cream
Spread the frosting immediately after beating with the powdered sugar, or it will start to set
Store frosted cake in an airtight container, refrigerated for up to 72 hours