Preparation

Make the dough

• In a stand mixer fitted with the paddle attachment, combine the flour, sugar, coconut flour, butter, and salt. Mix on low speed until the butter begins to break up, about 1 minute. Increase the speed to just below medium and continue beating until the mixture looks sandy with butter pieces the size of tiny pebbles, about 2 minutes more.
• Turn the mixer off and add the whole egg and egg yolk (reserve the remaining white). Mix on low speed until incorporated. Increase the speed to medium and mix just until the dough pulls away from the sides of the bowl, about 30 seconds. Do not overmix.
• Divide the dough in half, press each half into a disk about 4 inches across, and wrap in plastic. Refrigerate 1 disk for at least 3 hours and up to 24 hours. (Freeze the other disk for up to a month; thaw in the refrigerator before using.)

Shape and bake the crust

• Lightly flour a work surface and a rolling pin. Spray a 9-1/2- to 11-inch fluted tart pan with a removable bottom with cooking spray.
• Working quickly, roll the dough disk into a 1/8-inch-thick round. Transfer the dough to the tart pan and gently coax it into the pan, allowing the excess dough to hang over the sides. Roll the rolling pin over the top of the pan to cut off the excess dough. Patch any tears or cracks with the scraps. Refrigerate the crust for at least 20 minutes or freeze for 10 minutes. Meanwhile, position a rack in the center of the oven and heat the oven to 375°F.
• Line the crust with parchment paper and fill it to the very top with beans or pie weights. Bake until the edges are golden, 18 to 20 minutes. Carefully remove the weights and paper. Bake until the center of the crust looks dry and is just beginning to color, about 5 minutes.
• Beat the reserved egg white. Brush the inside of the crust with some of the egg white (you won’t need it all) and return the crust to the oven until the egg white has dried, about 2 minutes. (This prevents the crust from getting soggy once you add the filling.) Cool completely on a rack.
• Spread the mashed banana evenly over the interior of the cooled crust.

Make the pastry cream

• Combine the milk and 2 Tbs. of the sugar in a 3- to 4-quart saucepan. If using a vanilla bean, split it and scrape the seeds into the milk, then add the pod. Bring to a boil, stirring occasionally, about 5 minutes. Remove the vanilla bean pod, if using.
• Meanwhile, in a medium heatproof bowl, whisk the remaining 2 Tbs. sugar with the egg, egg yolk, and cornstarch until smooth, about 1 minute. Whisk half of the hot milk into the egg mixture until combined. Off the heat, whisk the contents of the bowl back into the remaining milk in the saucepan.
• Cook over medium heat, whisking constantly, until the mixture reaches a boil; it will thicken. Continue to cook for another minute past the boil, whisking constantly. Off the heat, whisk in the butter, vanilla paste or extract, if using, and the rum. Strain the pastry cream through a medium-mesh sieve into a medium heatproof bowl. Cover with plastic wrap pressed onto the surface of the pastry cream to prevent a skin from forming. Refrigerate until cold, at least 2 hours and up to 3 days.

Fill and top the tart

• In a medium bowl, beat the cream to stiff peaks with an electric hand mixer or a whisk.
• Beat the pastry cream to soften and smooth it. With a large spatula, fold the whipped cream into the softened pastry cream until thoroughly combined. Fill the crust with the crème légère, evening it out with the spatula. Refrigerate for at least 1 hour before topping with the fruit.
• Arrange the pineapple and mango on top of the filling. Note that because the filling is soft, you’ll need to work quickly while the filling is cold and more firm. Because it’s hard to move the fruit once it’s on the filling, you may want to practice your pattern on an empty sheet pan before placing it on the tart.
• Put the jelly and water or wine in a small saucepan. Simmer and whisk together to form a smooth glaze. Allow to cool slightly. Brush it on the exposed fruit, being careful not to pull up the cream filling from below. Garnish the tart with the coconut shavings. Refrigerate the tart until ready to serve. Serve the tart cold, within 6 hours of topping. Garnish with the cilantro leaves just before serving.