For the crust: Preheat oven to 350ºF. In food processor, combine crumbled cookies, flour, sugar and cinnamon. Process to fine crumbs. With machine running, drizzle melted butter through feeding tube until crumbs are evenly coated (mixture will not form a ball but remain crumbly).
Turn mixture out into a 9-inch pie pan. Using your fingers, evenly distribute crumbs over the bottom and up the sides of pan, gently pressing into an even thickness. Bake on a baking sheet until lightly browned, about 10 minutes. Cool completely on a rack.
For the filling: Whisk 2 cups of milk, the sugar and salt in a medium saucepan. Put over medium heat and bring to a boil. In medium heatproof bowl, whisk remaining 1/2 cup milk and cornstarch. Whisk in egg and egg yolks. Slowly whisk a third of the hot milk into the cornstarch mixture. As milk continues to boil, whisk warmed egg mixture back into milk, whisking constantly until mixture comes to a gentle boil. Whisk constantly for a couple of minutes until mixture thickens; then continue whisking and cook for 30 seconds more. Remove from heat and whisk in butter and vanilla.
Pour half of the hot filling into the piecrust. Layer on the sliced banana. Pour the remaining half of the filling over the banana, scraping the pan clean with a rubber spatula. Use the spatula to smooth over the top. Press plastic wrap directly onto the filling and refrigerate the pie until cold, at least 1 hour. Before serving, let the pie sit at room temperature for 30 minutes. Arrange sliced strawberries in a spoke pattern over the top just before serving.
Tip: If you like, serve each a slice with a small spoonful of whipped cream.