Creating the Velvety Caramel Custard Tart involves a few steps, but with careful attention, it becomes a straightforward process. Follow these instructions to achieve the perfect tart.

Step 1: Prepare the Tart Shell

1. Combine Dry Ingredients: In a large bowl, mix the flour, sugar, and salt.
2. Add Butter: Cut in the diced butter using a pastry blender or fork until the mixture resembles coarse crumbs.
3. Incorporate Egg Yolks: Add the egg yolk and mix until combined.
4. Add Water: Gradually add ice-cold water, one tablespoon at a time, until the dough holds together when pressed.
5. Chill the Dough: Shape the dough into a disc, wrap it in plastic wrap, and refrigerate for at least 30 minutes.
6. Roll Out the Dough: On a lightly floured surface, roll out the dough to fit a 9-inch tart pan. Carefully transfer the dough to the pan, pressing it into the edges, and trim excess.
7. Pre-bake the Shell: Preheat the oven to 350°F (175°C). Line the tart shell with parchment paper, fill with pie weights or dried beans, and bake for 15 minutes. Remove weights and parchment, and bake for another 10 minutes or until lightly golden. Let it cool.

Step 2: Make the Caramel Custard Filling

1. Cook Sugar: In a medium saucepan, combine the sugar and water over medium heat. Cook without stirring until the sugar melts and turns a deep amber color.
2. Add Cream: Remove from heat and carefully add heavy cream, stirring constantly. Be cautious as the mixture will bubble vigorously.
3. Add Vanilla: Stir in the vanilla extract and salt.
4. Mix Eggs: In a separate bowl, whisk together egg yolks and whole eggs until smooth.
5. Combine Mixtures: Slowly pour the caramel mixture into the eggs, whisking constantly to prevent the eggs from curdling.
6. Strain the Custard: Pour the mixture through a fine-mesh sieve into a clean bowl to remove any cooked egg bits.

Step 3: Assemble and Bake the Tart

1. Fill the Tart Shell: Pour the caramel custard filling into the pre-baked tart shell.
2. Bake: Reduce oven temperature to 325°F (160°C) and bake the tart for 25-30 minutes, or until the custard is set but still slightly wobbly in the center.
3. Cool: Allow the tart to cool to room temperature, then refrigerate for at least 2 hours before serving.