Périgord Walnut Tart

Name: Périgord Walnut Tart
Ingredients: 2 cups all-purpose flour; 2 tablespoons sugar; 1 teaspoon salt; 9 tablespoons cold unsalted butter, cut into 1/2-inch cubes; 1 large egg; 3 to 5 tablespoons ice water; 2 cups walnuts (8 ounces); 2/3 cup heavy cream; 1/2 cup water; 1 1/2 cups plus 2 tablespoons sugar, divided; 1/2 stick unsalted butter, cut into tablespoons; Equipment: a 9-inch tart pan with a removable bottom