Whisk the whole eggs and egg yolks in a medium heat-proof bowl, then set the bowl near the stove. Having the eggs ready makes tempering quick once the milk mixture is hot.

Whisk the sugar, cornstarch, and salt in a medium saucepan, then gradually whisk in the milk until smooth. Cook over medium-high heat for about 6 minutes, whisking constantly, until the mixture begins to bubble and thicken.

Slowly stream about 1 cup of the hot milk mixture into the eggs while whisking continuously. The gradual addition warms the eggs gently and helps prevent scrambled bits.

Pour the warmed egg mixture back into the saucepan while whisking, then return the pan to medium heat. Keep the whisk moving so the custard stays smooth across the bottom and sides of the pan.

Bring the custard back to a bubble and continue whisking for 6 to 8 minutes. It is ready when it looks glossy and thick and the whisk leaves a distinct track across the surface.

Remove the pan from the heat and whisk in the cold butter and vanilla until completely smooth. The custard may look thicker at first and then loosen slightly as the butter melts, which is normal.

Transfer the custard to a bowl and press plastic wrap directly against its surface. Let it cool at room temperature for 2 to 3 hours, or refrigerate after 30 minutes to speed the cooling.

Roll the pie dough on a lightly floured surface into a 13-inch circle, turning it as you work. Roll from the center outward and add flour only as needed to keep the dough from sticking.

Fit the dough into a 9-inch pie pan without pulling or stretching it, then trim it to leave a 1-inch overhang. Fold the overhang under, crimp the edge, and prick the bottom all over with a fork.

Chill the shaped crust for 1 hour, or freeze it for 30 minutes. Cold dough holds its crimp better and is less likely to shrink while baking.

Heat the oven to 425°F while the crust chills. The oven should be fully hot before the chilled dough goes in for a crisp start.

Line the chilled crust with foil or parchment and fill it with baking beans, then bake for 15 to 25 minutes. The edges should turn golden and the bottom should look dry and matte when the weights come out.

Whisk the egg and heavy cream together in a small bowl. This mixture will create a thin seal over the partially baked pastry.

Lift out the lining and baking beans, brush the crust with the egg wash, and shield the rim with foil if it is browning quickly. Bake for another 10 to 12 minutes, until the entire crust is golden, then cool it completely.

Spread 1 cup of cooled custard evenly across the bottom of the completely cooled pie shell. The custard should resist the spatula slightly instead of flowing like a sauce.

Peel and cut two bananas into ½-inch slices, then arrange them over the custard layer. Overlap the slices lightly so each piece stays in place when the pie is cut.

Spoon the remaining custard over the bananas and smooth the top, making sure all fruit is covered. Cover the pie and refrigerate for at least 6 hours, until the filling feels firm when the pan is gently nudged.

Beat the heavy whipping cream, confectioners' sugar, and vanilla on medium speed for about 2 minutes. Stop at soft peaks, when the cream holds a curl that gently folds over at the tip.

Dollop or pipe the whipped cream over the chilled pie, then slice the remaining 1 to 2 bananas for garnish if desired. Add the fresh banana slices immediately before serving to keep the top looking bright.