Preheat oven to 350°F (175°C). Lightly butter the sides of a 7-inch tart pan (or line the base of a cake pan with parchment for easy lifting).
Make the crumb crust: Stir cookie crumbs, sugar, salt, and optional cinnamon. Drizzle in melted butter and mix until it resembles damp sand and holds a clump when squeezed. Press firmly into the pan across the base and up the sides using a flat-bottomed cup. Bake 8–10 minutes until fragrant and slightly darker, then cool completely.
Infuse the milk: In a saucepan, combine milk, cinnamon stick, and half the vanilla. Heat over medium until steaming with tiny bubbles at the edge (2–3 minutes); do not boil. Cover, remove from heat, and steep 10 minutes. Discard cinnamon stick, then rewarm the milk until steaming again before tempering.
Build the custard base: In a bowl, whisk together sugar, flour, cornstarch, and salt. Whisk in the egg and yolk until completely smooth.
Temper and thicken: Slowly pour the hot milk into the egg mixture while whisking constantly. Return mixture to the saucepan and cook over medium-low, scraping the bottom and corners with a spatula, until glossy and thickened (3–5 minutes) or 180–185°F (82–85°C). If lumpy, whisk vigorously off heat and/or strain through a fine-mesh sieve.
Finish the custard: Off heat, whisk in butter and remaining vanilla. Custard should be thick but pourable, like soft pudding.
Fill and chill: Strain hot custard into the cooled crust. Tap the pan to release bubbles and smooth the top. Cool 15 minutes at room temperature, then refrigerate at least 3–4 hours (or overnight) until set with a gentle wobble.
Finish and slice: Mix cinnamon and sugar and sift generously over the tart. Remove tart ring. Slice with a thin knife warmed under hot water and wiped dry between cuts for clean edges.