Preheat your oven to 350 degrees F. Spray three 8-inch or four 6-inch round cake pans with cooking spray, line with parchment rounds, then spray the parchment. Set aside.
In a medium bowl, sift the flour, baking powder, baking soda and salt. Set aside.
Combine the milk and greek yogurt in a small bowl and set aside.
In a stand mixer fitted with the paddle attachment, beat the butter, oil, and sugars on medium-high speed until pale and fluffy, about 5 minutes. Lower the speed to medium and add the eggs, one at a time, mixing until each is fully incorporated before adding the next (about 30 seconds). Add the vanilla.
With the mixer on low, add the dry ingredients alternating with the milk and yogurt mixture into the creamed mixture, starting and ending with the dry ingredients. Mix until just incorporated and remove bowl from stand mixer. Using a large stainless steel whisk, stir/whisk the mashed banana into the mixture--you can use use some strength and blend well, ensuring all of the ingredients are well mixed. (If you don't have a large stainless whisk, you can fold the banana mixture in using a rubber spatula until incorporated.)
Before pouring the batter in the pans, divide the sliced bananas into TWO of the pans, leaving one pan empty. After arranging the banana slices on the bottom of the pan, pour a few tablespoons of the caramel sauce over the bananas, leaving one pan without bananas or caramel sauce.
Once the bananas and caramel sauce are in the pans, divide the batter evenly between the three prepared pans, using a kitchen scale for the best accuracy. (For this recipe, you'll have about 1.4 lbs of batter in each pan).
Carefully smooth the batter with small offset palette knife, and bake until a toothpick or skewer comes out clean, about 28 to 33 minutes. The cake layer without the bananas and sauce will take less time to cook. I took this layer out of the oven at 28 minutes. The two layers with bananas and sauce took 33 minutes to cook.
Let pans cool on wire rack for 10 minutes, then invert cakes onto racks, gently, peeling away parchment rounds. Let cool completely.
When the cakes are completely cooled, level the one cake layer that doesn't have bananas and sauce on it. Using a pastry brush, soak the cake with the remaining caramel sauce.
The cake layers can be made up to a week in advance, but I just making this cake as close to serving time as possible to keep the bananas looking fresh. If you plan to make it in advance, cover each cake layer with two layers of plastic wrap. When you're ready to frost your cake, take the cake layers out of the freezer an hour before you decorate.