Make the Pudding: In a medium bowl, whisk together the whole eggs and egg yolks. Set aside. Add sugar, cornstarch, and salt to a medium pot. Whisk to combine, then pour in milk and whisk again to remove all clumps.

Place over medium-high heat and cook, whisking constantly, until just starting to bubble and thicken, about 6 minutes. Remove from heat.

Temper the egg mixture by slowly drizzling about 1 cup of the hot milk mixture into the eggs while whisking vigorously. Slowly pour the tempered egg mixture back into the pot while whisking.

Return pot to medium heat and cook, whisking constantly, until mixture bubbles again. Continue cooking until thick enough that the whisk leaves clear traces, 6-8 minutes.

Remove from heat. Add butter and vanilla, whisking until butter is completely melted. Transfer to a bowl and press plastic wrap directly onto surface. Cool to room temperature, 2-3 hours.

Prepare the Crust: Roll pie dough into a 13-inch circle on lightly floured surface. Transfer to 9-inch pie pan without stretching. Trim excess, leaving 1-inch overhang. Fold edges under and crimp.

Prick bottom all over with fork. Refrigerate 1 hour or freeze 30 minutes. Preheat oven to 425°F.

Line chilled crust with foil and fill with pie weights. Bake 15-25 minutes until edges are golden and bottom appears dry.

Meanwhile, whisk together egg and heavy cream for egg wash. Remove weights and foil, brush crust with egg wash. Bake another 10-12 minutes until golden all over. Cool completely.

Assemble the Pie: Spread 1 cup cooled pudding in bottom of pie shell. Slice 2 bananas into 1/2-inch pieces and layer over pudding.

Spoon remaining pudding over bananas and smooth top. Cover and refrigerate at least 6 hours.

Add Topping: When ready to serve, beat heavy cream, powdered sugar, and vanilla to soft peaks. Spread over pie. Garnish with remaining sliced bananas if desired.