Heat the butter in a small saucepan over medium heat, whisk frequently. Continue to cook until melted. The butter will foam and then subside. Cook until brown bits develop in the bottom of the pan and the butter turns a light golden color, about 8 minutes. Remove from heat.
Line a small bowl with foil, letting some excess to hang over the sides. Pour the brown butter into the bowl, leaving behind the dark bits in the bottom of the pot. Freeze for 1 hour.
Remove the solidified brown butter from the bowl and cut into small chunks.
Combine both flours, salt, and sugar (if using) in a large bowl.
Add the eggs and carefully cut them into the flour with a silicone spatula.
Using a pastry blender, cut the butter into the flour mixture until the butter pieces are smaller than hazelnuts but larger than peas, about 2 minutes.
Fold in the water, one tablespoon at a time until no dry crumbs remain in the bottom of the bowl. Pick up a chunk of dough and squeeze it together. If it doesn't come together, add 1-2 tablespoons more water and cut it in.
Turn the dough out onto a lightly flour surface. Knead the dough by smearing it with the heel of your hand just long enough to get it to come together.
Divide the dough in half and flatten each half into a disk. Wrap the disks tightly in plastic wrap and refrigerate for at least 2 hours.
Make the filling
Heat the butter in a small saucepan over medium heat, whisk frequently. Continue to cook until melted. The butter will foam and then subside. Cook until brown bits develop in the bottom of the pan and the butter turns a light golden color, about 8 minutes. Remove from heat and set aside.
Whisk the granulated sugar, egg yolks, salt, and cornstarch together in large bowl until smooth.
Heat the milk and heavy cream in a medium saucepan over medium-low heat until steaming and bubble begin to form around the edges.
While whisking continuously, gradually add the hot milk mixture to the egg mixture. After all the milk has been added, return the mixture to the saucepan and cook over medium heat, whisking constantly, until thickened to the consistency of warm pudding, 4 to 6 minutes.
Remove from heat and whisk in the brown butter and vanilla.
Press the custard through a fine-mesh sieve into a bowl. Place plastic wrap directly onto the surface of the custard to prevent a skin from forming. Refrigerate for at least 2 hours.
On a lightly floured surface roll 1 disc of pie dough into 12-inch round. (You can freeze the second disc to use at a later time.) Place in a 9-inch pie plate and trim off any excess along the edges, leaving a 1-inch overhang. Fold the excess under so it sits up on the rim of the pie dish. Flute the edges with fingertips. Refrigerate for 30 minutes.
Adjust oven rack to lower third position and heat oven to 375°F.
Line the crust with parchment paper and fill with pie weights. (Make sure the parchment paper is large enough to be lifted out easily.) Bake for 20 minutes. Carefully remove the parchment paper and weights, rotate the pie, and continue baking until the crust is golden brown, 10-15 minutes. Let cool to room temperature.
Peel and slice the bananas about 1/4- inch thick.
Arrange half the bananas over the bottom of the crust. Spread half of the chilled custard over the bananas. Arrange the remaining bananas over the custard and top with the remaining custard.