In a medium heat-proof bowl, whisk together the whole eggs and egg yolks. Set aside.

Add the sugar, cornstarch, and salt to a medium pot. Whisk to combine then pour in milk and whisk again to distribute all the dry ingredients. You don't want clumps! Place over medium-high heat and cook, whisking constantly, until just starting to bubble and thicken, about 6 minutes. Remove from the heat.

Temper the egg mixture by drizzling about 1 cup of the hot milk mixture into the egg mixture while whisking vigorously. Slowly pour the tempered egg mixture back into the pot, while whisking. Return the pot to medium heat.

Cook, gently whisking constantly, until the mixture starts to bubble again. Continue cooking until the mixture is thick enough that the whisk leaves a trace when swooped through it, 6 to 8 minutes. Remove from the heat.

Add the butter and vanilla and whisk until the butter is fully melted and well combined. (The pudding may thicken up more initially when the cold butter is added in, and then look runnier after it is melted and combined. This is normal!) Transfer the pudding to a bowl and press plastic wrap onto the surface of the filling. Let it cool to room temperature, 2 to 3 hours. (Once it has cooled for 30 minutes, you can place it in the refrigerator to speed this up.)

On a lightly floured surface, roll the pie crust into a 13-inch circle. (Roll from the center to the edge, turning and flouring the dough so it doesn't stick to the board.)

Transfer the dough to a 9-inch pie pan, trying not to stretch it as you transfer it. (Stretching can cause shrinkage when baked.) Press the dough into the pie pan, and trim the excess dough leaving a 1-inch overhang around the pan. Fold the edge under and crimp with either your fingers or the tines of a fork. Prick the crust all over the bottom with the tines of a fork. Refrigerate the crust for 1 hour, or freeze for 30 minutes.

While the crust chills, preheat the oven to 425°F.

Line the inside of the chilled pie crust with a piece of foil or parchment paper. Fill with baking beans. Bake for 15 to 25 minutes until the edges are golden brown, and the inside bottom of the crust appears dry on the surface.

Meanwhile, in a small bowl, whisk together the egg and heavy cream.

Carefully remove the foil with the baking beans and brush the crust all over with the egg wash. Tent just the edges of the crust with foil if needed. Bake for another 10 to 12 minutes, or until the crust is golden all over. Let the crust cool completely.

Add 1 cup of cooled pudding onto the bottom of the cooled pie shell, and spread it into an even layer with a spatula.

Peel and slice two bananas into ½-inch thick pieces. Scatter them over the pudding, shingling them slightly if needed. Spoon the remaining filling into the pie shell over the bananas, and smooth out the top. Cover and refrigerate for at least 6 hours.

When you’re ready to serve, combine the heavy cream, powdered sugar, and vanilla in the bowl of a stand mixer (or a large mixing bowl with a handheld electric mixer). Beat on medium speed until you reach soft peaks, about 2 minutes.

Dollop or pipe the whipped cream over top of the pie. You can chill the pie for another hour before serving if needed. To garnish, peel and slice the remaining 1 to 2 bananas and decorate the top with the banana slices if desired.