How to Make It
Preheat the oven to 350°. Line a 9-by-13-inch baking pan with parchment paper, leaving a 2-inch overhang on the short sides. In a large bowl, using a handheld mixer at low speed, cream the butter. Beat in the confectioners' sugar. Add the whole eggs and beat until incorporated, then beat in the flour and salt. Press the pastry into the prepared pan in an even layer, 1/4 inch thick. Freeze until firm, 10 minutes.
Line the pastry with parchment paper and fill with pie weights. Bake for 35 minutes, until just set. Carefully remove the pie weights and parchment. Brush the shell with the egg white and bake for 20 minutes longer, until golden and cooked through. Let cool.
In a saucepan, bring the cream, vanilla bean and seeds to a simmer. Cover; keep warm.
In a large, heavy saucepan, stir the sugar into 1/4 cup of water. Simmer over moderate heat, without stirring, until a deep amber caramel forms, 7 minutes.
Remove the caramel from the heat and carefully add the cream. When the bubbling subsides, stir in the butter. Insert a candy thermometer and cook over moderately high heat, stirring with a wooden spoon, until the caramel reaches 240°, 10 minutes. Discard the vanilla bean and stir in the salt. Pour the caramel over the shell. Refrigerate until firm, 4 hours or overnight; bring to room temperature. Remove the bar from the pan using the parchment overhang; cut into squares.