Prepare the Tart Crust: Combine flour, granulated sugar, and salt in bowl of a food processor. Pulse until well combined, 5 to 7 times. Add butter, and pulse just until mixture resembles coarse sand, 5 to 7 times.
Add egg yolk and 2 tablespoons ice water, and pulse until mixture comes together and forms a ball, adding remaining 1 tablespoon ice water, 1 teaspoon at a time, if necessary. Shape and flatten dough into a disk. Wrap in plastic wrap, and chill 1 hour or up to 2 days.
Grease (with butter) a 9-inch fluted tart pan with removable bottom. Unwrap chilled dough disk, and place on a lightly floured surface. Let stand at room temperature until slightly softened, about 5 minutes. Sprinkle dough with flour, and roll into a 10-inch circle. Gently drape dough circle over prepared tart pan; place in pan. Press dough on bottom and up sides of pan. Cover and chill 30 minutes or up to 2 days.
Prepare the Caramel Filling: Preheat oven to 350°F. Stir together brown sugar, flour, and salt in a large bowl.
Whisk together whole milk, dark corn syrup, melted butter, vanilla extract, and egg yolk in a bowl, and pour over brown sugar mixture. Stir until well combined.
Whisk egg white in a medium bowl until thick and creamy and holds soft peaks. Fold beaten egg white into corn syrup mixture until incorporated and mixture becomes caramel colored and smooth. Spoon Caramel Filling into prepared Tart Crust. Place pan on a rimmed baking sheet.
Bake in preheated oven until center is set and filling is puffed and golden brown, 45 to 50 minutes. Transfer to a wire rack, and cool completely, about 3 hours.
Prepare the Brandy Whipped Cream: Using chilled beaters and a large chilled bowl, beat cold whipping cream with an electric mixer on high speed until thickened, about 2 minutes. Add sugar, and beat until stiff