Preheat oven to 325F. Line 2 9" (or 3 8") cake pans with parchment paper.
Sift the flour, baking powder 1 1/4 cup (250 g/9 oz) sugar and salt twice.
In a large bowl, whisk together the egg yolks, milk , oil ,vanilla and banana puree to combine.
Slowly add the flour mixture and stir until smooth.
In a bowl of a standing mixer, fitted with the whisk attachment, beat the egg whites,cream of tartar and the remaining of the 1/4 cup (50 g/2 oz) sugar.
Whip until stiff meringue is form.
Using a rubber spatula, gently fold 1/4 of the meringue into the batter. then fold the remaining of the meringue.
Divide the batter between the cake pans and bake until the cakes springs back when touched in the middle. About 35-40 minutes.
Allow the cakes to cool to room temperature before removing from the pans.
Slowly add the milk to the flour, 3-4 TBS at a time and whisk to combine between each addition.
You should get a smooth sticky looking mix.
Place the flour and milk mixture in a heavy bottom saucepan over medium heat.
Allow the mixture to simmer while stirring.
Once the mixture is simmering, vigorously whisk with a hand whisk for about 30-45 seconds more. The mixture should be thick and pasty.
Remove from heat and place a plastic wrap on top. Allow to cool to room temperature.
Place the butter and sugar in a bowl of a standing mixer and mix on high for about 5 minutes.
Add the cool flour and milk mixture and continue mixing for an additional 5 minutes or until you get a smooth and airy buttercream.
Add the Cookie Butter spread and mix until combine.
Place the sugar and water in a heavy bottom saucepan and allow to boil over medium heat.
Divide each cake into two layers. Place one layer on top of a cake board or on top of a serving plate.
Brush the cake with simple syrup and spread with 1/4 cup of buttercream.
Repeat the above steps but spread plain cookie butter spread on the second layer.
frost the cake with the remaining butter cream and decorate with cookies and dried bananas.