Step 1

Heat the oven to 375°F with a rack in the middle position. In a medium bowl, whisk together the flour and ¼ teaspoon of the salt. Make a well in the center; set aside.

Step 2

In a small saucepan, combine the butter, shortening and ¼ cup water. Bring to a simmer over medium-high, stirring to melt the solids. As soon as they’re melted and the mixture is simmering, pour it into the well of the dry ingredients. Working quickly, stir with a silicone spatula until the dry ingredients are evenly moistened and without any dry patches; the dough will be very soft and resemble wet mashed potatoes. Turn it out onto a large sheet of plastic wrap and, using your hands, form it into a 6- to 8-inch disk.

Step 3

Cover the dough disk with another large sheet of plastic wrap and roll it into a 12-inch round of even thickness. Peel off the top sheet of plastic. Using the bottom sheet of plastic, carefully flip the round into a 9-inch pie plate, centering it as best you can. Ease the dough, still on the plastic, into the corners and up the sides of the pie plate. Carefully peel off the plastic. If needed, patch any tears in the dough. Trim the excess dough and flute or crimp the edge. Carefully line the dough with a large sheet of foil, gently pressing it into the corners and up the sides, then fill with about 2 cups pie weights.

Step 4

Bake until the dough is set, about 20 minutes. Carefully lift out the foil with weights, then prick the pie shell all over with a fork to deflate any air bubbles and prevent additional ones from forming. Bake until the shell is lightly browned, another 12 to 15 minutes. Transfer to a wire rack to cool until barely warm to the touch. Reduce the oven to 325°F.

Step 5

Peel the bananas and slice them into ¼-inch rounds. Lay as many slices in the pie shell as will fit in a single, tightly packed layer, then set the pie plate on a rimmed baking sheet. Add the remaining banana slices to a blender, along with the whole eggs plus yolk, milk, condensed milk, vanilla and the remaining ¼ teaspoon salt. Blend until smooth, 15 to 30 seconds.

Step 6

Pour the mixture into the pie shell, taking care not to overfill it (some of the banana slices will rise to the surface); the pie shell may not hold all of the filling, depending on how much it shrank during prebaking. Carefully transfer the baking sheet to the oven and bake until the filling is puffed and lightly browned at the edges and the filling jiggles only slightly when the pie plate is gently shaken, 55 to 65 minutes. Transfer to a wire rack and cool to room temperature.

Step 7

If caramelizing the surface, sprinkle the sugar evenly onto the cooled pie. Using a kitchen torch, caramelize the sugar until spotty brown. Serve within an hour, before the sugar crust softens. (The pie also is good served chilled, but if caramelizing the sugar, do so just before serving, as refrigeration will soften the sugar crust.)