Crust: In a food processor, pulse brown sugar and 1/4 cup flour to combine. Add salt and remaining 1 cup flour; pulse to combine. Add butter and pulse just until mixture forms peasize clumps. Add egg yolk and ice water and pulse just until mixture starts to come together. Turn out dough onto a piece of plastic wrap and bring together in plastic to form a disk. Wrap and refrigerate until firm but still pliable, about 30 minutes.
On a lightly floured surface, roll out dough to a 13-inch round. Press into bottom and up sides of a 10-inch fluted tart pan with a removable bottom. Trim edges flush with rim. Refrigerate until firm, about 45 minutes.
Preheat oven to 375 degrees. Line dough with parchment; fill with dried beans or pie weights. Bake 25 minutes, then remove parchment with beans. Bake until crust is golden and set throughout, about 10 minutes more. Let cool completely on a wire rack.
Filling: In a medium saucepan, heat milk and cream with vanilla pod and seeds until almost boiling. Meanwhile, in a bowl, whisk together egg and granulated sugar until combined and thick, then whisk in flour, cornstarch, and salt. Reduce heat to medium, then slowly whisk egg mixture into cream mixture. Continue whisking until bubbles appear in center of pan, about 3 minutes.
Remove from heat and stir in butter until melted. Pass through a fine sieve into a bowl; discard solids. Pour filling into cooled crust. Sprinkle with cinnamon and refrigerate at least 3 hours and up to overnight. Cut into wedges to serve.