In a food processor, pulse flour, confectioners' sugar, and cold butter until crumbly. Add egg yolk and ice water, pulsing until dough forms. Wrap and chill for 1 hour.
Roll dough to ¼-inch thickness, press into a 9-inch tart pan, and blind bake with weights at 375°F (190°C) for 15 minutes. Remove weights, bake 10 more minutes until golden. Cool.
For the caramel, boil granulated sugar and water in a saucepan until deep amber. Remove from heat, carefully whisk in warm heavy cream, then whisk in butter and salt.
Pour warm caramel into the cooled tart shell and chill for 30 minutes to set.
For the custard, heat milk, heavy cream, and vanilla bean paste until simmering. Let steep for 10 minutes.
Whisk egg yolks, sugar, and cornstarch. Slowly temper by whisking in half the hot milk mixture, then return everything to the saucepan.
Cook over medium-low heat, whisking constantly until thickened and boiling for 1 minute. Remove from heat and whisk in softened butter.
Pour hot custard over the set caramel layer. Cover surface with plastic wrap and chill for at least 4 hours before serving.
