For the cookies:
1/3 cup unsalted butter, at room temperature
1/3 cup shortening
1 cup powdered sugar, sifted
2/3 cup sugar
2 extra-large eggs, at room temperature
1 teaspoon pure vanilla extract
1 1/2 tablespoons water
3 cups cake flour, sifted
1 tablespoon baking powder
1/2 teaspoon kosher salt
1/2 teaspoon sea salt flakes
For the banana purée:
2 medium overripe bananas
2 tablespoons sugar
1/4 cup water
1 tablespoon freshly squeezed lemon juice
For the frosting:
1 cup (8 ounces) unsalted butter, at room temperature
1 cup powdered sugar, sifted
1/2 vanilla bean, seeds only
1 tablespoon pure vanilla extract
1/2 teaspoon kosher salt
1/2 teaspoon sea salt flakes
1/2 cup caramel sauce, store-bought or homemade, at room temperature
To finish:
8 ounces bittersweet chocolate (preferably 64% cacao), melted
2 ounces Valrhona Dulcey, melted, or caramelized white chocolate (see Recipe Note)
Make the Cookies: In the bowl of a stand mixer fitted with the paddle attachment, mix the butter and shortening on medium speed for 5 to 10 seconds. Add the sugars and mix on low speed to incorporate. Cream the mixture on medium speed until it is aerated and looks like frosting, 3 to 4 minutes. Scrape the sides and bottom of the bowl with a rubber spatula to bring the batter together.
Crack the eggs into a small cup or bowl, and add the vanilla and water.
In a bowl, whisk together the flour, baking powder, and salts.
On medium speed, add the eggs and vanilla, one egg at a time, mixing the first briefly before adding the second, until the batter resembles cottage cheese, approximately 5 seconds for each egg. Scrape the sides and bottom of the bowl with a rubber spatula to bring the batter together. Mix on medium speed for 20 to 30 seconds to make nearly homogeneous.
Add the flour mixture all at once and mix on low speed until the dough just comes together but still looks shaggy, approximately 1 minute. Do not overmix. Remove the bowl from the stand mixer. With a plastic bench scraper, bring the dough completely together by hand.
Stretch out a long sheet of plastic wrap on a work surface and put the dough on top. Pat into a rectangle, wrap tightly, and refrigerate until chilled throughout, at least 2 hours or preferably overnight.
Unwrap the dough, place on a work surface lightly dusted with flour, and cut in half. Leave at room tempe