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 powdered sugar. Add 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 (or fill with dry beans). 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 heavy saucepan, bring the cream, vanilla bean and seeds to a simmer. Remove from heat, cover and keep warm.

In a large, heavy-bottomed saucepan, stir the sugar into 1/4 cup of water. Simmer over moderate heat, without stirring, until deep amber caramel forms, about 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°, about 10 minutes. Discard the vanilla bean and stir in the salt.

Immediately pour the caramel over the shell. Refrigerate until firm for 4 hours or overnight; bring to room temperature. Remove the bars from the pan using the parchment overhang; cut into squares.

When bars have cooled about 30 minutes in the refrigerator, I sprinkled them lightly with a little extra kosher salt.