To make the dough: In a large bowl, beat the butter, shortening, and sugars at medium speed until fluffy., Crack the eggs into a small bowl and add the vanilla and water., In a separate bowl, whisk together the flour, baking powder, and salts., Add the egg mixture to the butter/sugar mixture a third at a time, mixing until the batter resembles cottage cheese. Scrap down the bowl and mix on medium speed for another 30 seconds., Add the flour mixture all at once and mix on low speed until the dough just comes together. Divide the dough in half, pat each half into a rectangle, wrap in plastic, and refrigerate at least 2 hours, or overnight., Preheat the oven to 300°F. Line two baking sheets with parchment., On a lightly floured work surface, roll one piece of dough 1/4" thick. Prick the dough all over with a fork. Cut out cookies using a 2" round cutter, and place on the prepared pans., Bake one pan at a time for 15 to 17 minutes, until the edges of the cookies are lightly browned. Remove from the oven and let the cookies cool completely on the pan. Repeat with the remaining dough., To make the banana purée: In a small heavy-bottomed pan set over medium heat, cook the bananas, sugar, water, and lemon juice until the bananas are soft enough to fall apart when nudged with a spoon, 5 to 7 minutes. Purée with an immersion blender or food processor until smooth. Transfer to a bowl, place plastic wrap on the surface, and refrigerate until chilled., For the filling: In the bowl of a stand mixer fitted with the paddle attachment, beat the butter with the sugar until the mixture is aerated and pale in color. Scrape the sides and bottom of the bowl. Mix in the vanilla bean seeds, vanilla, and salts., Warm the butterscotch sauce and mix it with the banana purée. Make sure this mixture is at room temperature. Fold the banana-butterscotch mixture into the rest of the filling until evenly mixed. If the mixture curdles, whip it while warming the outside of the bowl gently with a hair dryer until smooth., To assemble the cookies: Place the filling in a pastry bag fitted with a round tip. Make pairs of similar-sized cookies, then turn half of them over. Pipe a "bubble" of filling on the bottoms of the overturned cookies, leaving a 1/4" border clear at the edge. Top each filled cookie with a second cookie and press lightly to adhere. Put the cookies on a parchment-lined baking sheet and refrigerate until the filling has set, approximately 30 minutes., To finish the cookies: Dip the top of each sandwich cookie in the melted bittersweet chocolate, and place chocolate side up on the parchment-lined pans. Drizzle the tops with melted white chocolate.