Preheat oven to 350°. In a bowl, with a mixer on medium speed, beat 1/2 cup butter and the brown sugar until smooth. Beat in egg yolk and vanilla. Add flour and 1/4 teaspoon salt and beat until mixture forms a ball.
Press dough over bottom and up sides to rim of a 9-inch fluted tart pan with removable rim. Freeze until firm, 15 minutes.
Meanwhile, spread walnuts in a large baking pan. Bake until lightly golden under skins, about 8 minutes (leave oven on). When walnuts are cool enough to handle, chop 2 cups; reserve remainder.
Bake tart shell until golden, 16 to 18 minutes. Transfer to a wire rack.
Meanwhile, in a 3- to 4-quart pan over medium heat, combine granulated sugar, corn syrup, and 1/4 cup water. Stir until sugar is dissolved, then increase heat to high and boil, swirling mixture occasionally, until mixture is a deep golden brown, 10 to 15 minutes. Remove from heat and stir in crème fraîche, remaining 1/4 cup butter, and remaining 1/2 teaspoon salt (mixture will foam). Stir until smooth.
Spread chopped walnuts in tart shell, then pour in hot caramel; spread level. Garnish edge of tart with reserved walnut halves. Let tart cool until caramel is firm enough to slice, at least 4 hours, or chill up to 2 days (bring to room temperature before serving). Cut into thin wedges.