Place the sugar in a large, dry, light-colored skillet over medium heat. Let the sugar melt, shaking the pan and whisking frequently. It will turn golden as it melts. As soon as it is melted and smooth, remove from heat.
Add the softened butter, whisking, one tablespoon at a time until smooth. Slowly add the cream, whisking constantly. The caramel may start to seize and get lumpy as you add the cream, but this is no problem. Keep whisking and the lumps should melt. If not, they will melt in the next step anyway.
Return the pan to medium heat and cook the caramel, whisking frequently, until it starts to thicken slightly, about 5 minutes. Remove from heat and whisk in the vanilla and sea salt. Set aside to cool completely before using. You can put it in the refrigerator to speed this up.
Banana Cake
Preheat the oven to 350 F. Grease a 9 x 13 inch baking pan with butter or cooking spray. If you plan to unmold the cake, line the bottom of the pan with parchment paper for easy removal.
In a medium bowl, mash the bananas with a fork until smooth. You can use and immersion blender or electric beater to make the job easier. Set bananas aside.
In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and brown and white sugars on medium speed, until fluffy, about 3 minutes.
Beat in the eggs and vanilla until well combined, scraping down the sides of the bowl as need. Beat in the bananas.
Reduce the speed to low. Starting with the dry ingredients, alternate adding the flour mixture and buttermilk in 3 instalments each, being careful to only beat until just incorporated. The batter will be lumpy.
Pour the batter into the prepared pan. Drizzle 1/2 cup of the caramel sauce over the batter and swirl around with a butter knife. Bake in the preheated oven for 35-45 minutes, until a toothpick inserted in the center comes out clean. Set pan on a wire rack to cool completely, about 90 minutes or so.
Caramel Cream Cheese Frosting
In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter and cream cheese on medium speed until smooth.
Lower the speed to low, and beat in the icing sugar, one cup at a time until smooth, scraping down the sides of the bowl as needed.
Add a 1/2 cup of the caramel, along with the vanilla and salt, and beat until smooth and incorporated.
Frost the cooled cake, and drizzle with additional caramel (if the caramel is too thick to drizzle, heat a bit up in the microwave for a couple of seconds.