Make the pastry: Pulse flour, icing sugar, and salt in a food processor. Add cold butter and blitz until mixture resembles coarse sand.
Add egg yolk and 1 tablespoon ice water. Pulse and add more water until dough clumps. Form into disc, wrap, and chill 1 hour.
Roll dough to 3mm thickness between parchment. Fit into 9-inch tart pan, trim excess, and prick base. Freeze 20 minutes.
Preheat oven to 350°F (175°C). Line pastry with parchment and pie weights. Blind bake 20 minutes, remove weights, bake 10-12 minutes more until golden. Cool completely.
Make custard: Heat milk, cream, and vanilla bean to simmer. Remove from heat and steep 15 minutes.
Whisk egg yolks, whole eggs, sugar, cornstarch, and salt until pale, about 2 minutes.
Remove vanilla pod. Temper eggs by slowly whisking in half the hot milk. Return to saucepan and cook over medium-low, stirring constantly, until custard coats spoon (170°F), 5-7 minutes.
Strain custard through fine-mesh sieve. Pour into cooled pastry shell. Reduce oven to 300°F (150°C).
Bake tart 35-45 minutes until edges set but center wobbles gently. Cool completely, then refrigerate 4 hours or overnight.
Make caramel: Cook sugar and water over medium-high without stirring until deep amber, 6-8 minutes. Remove from heat and whisk in cream and butter.
Cool caramel 5 minutes. Pour over chilled tart and tilt to spread. Set 15-20 minutes before slicing with warm knife.
