In the bowl of an electric mixer fitted with the paddle attachment, beat together the oil, eggs, granulated sugar, brown sugar, and vanilla on medium speed for 2 minutes. Scrape down the sides of the bowl with a rubber spatula. Add the banana puree and mix on low speed for 30 seconds. Scrape down the bowl again. With the mixer running on low speed, gradually add the flour mixture for a total mixing time of
30 seconds. Carefully scrape down the bowl. Refrigerate the batter for 30 minutes.
Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.
Using a pastry bag fitted with a large plain tip (the hole should be about as large as the tip of your ring finger), pipe 2-inch-diameter circles onto the prepared baking sheet, leaving 1 1/2 inches between them. Be careful to keep your pastry bag completely vertical to achieve nice circles. (If you have to wait between batches of whoopie pies, keep the batter refrigerated.)
Immediately put the baking sheet in the oven. Bake for 10 minutes without rotating the baking sheet. The whoopie pies are done when they are slightly golden in color and spring back when touched. Let the whoopie pies cool completely.
To fill the whoopies, turn half of them over so that they are bottomside up. Using a pastry bag, pipe a small dollop of the dulce de leche filling onto each whoopie bottom. Top with the remaining whoopies.
In the bowl of an electric mixer fitted with the paddle attachment, beat together the cream cheese and butter on medium speed until the mixture is light and fluffy, about 3 minutes. Scrape down the sides of the bowl with a rubber spatula. Add the vanilla, salt, and dulce de leche, and mix for 30 seconds on low speed.
With the mixer running on low speed, gradually add the confectioners' sugar, and then beat for 1 minute. Scrape down the bowl. If the filling is too soft to hold its shape, add more confectioners' sugar, a tablespoon at a time, until you reach the desired consistency. Store in the refrigerator, in an airtight container with plastic wrap pressed onto the surface of the filling, for up to 5 days.
Makes 2 cups