Preheat oven to 375˚F. On a lightly floured surface, roll the pie dough into a 12-inch round (see note). Gently transfer to a 9-inch pie plate. Trim the edges and crimp with your fingers or a fork. Place in the refrigerator for 30 minutes. Use the tines of a fork to prick holes into the dough all over. Line the dough with a sheet of parchment paper, and fill the pie shell with pie weights or dried beans. Bake for 15 minutes. Remove the pie weights and the paper. Set aside to cool.

Melt the butter in a medium skillet over medium heat. Cook 20-22 minutes, until the butter is dark, golden brown with dark brown flecks, and has a nutty aroma. Remove from heat and set aside to cool. In a large bowl, whisk together the brown sugar and eggs until fluffy and pale. Stir in the browned butter, then add the corn syrup, 1 1/2 teaspoons of vanilla, and salt, and whisk until smooth. Fold in the walnuts, then pour the filling into the pie shell. Use a rubber spatula to smooth the filling into an even layer. Return to the oven and bake about 40 minutes, until the filling is just set in the center. If the pie crust begins to get too brown, cover the edges with aluminum foil.

While the pie is in the oven, make the topping. Whip the heavy cream and sugar together in a medium bowl until stiff peaks form. Fold in the sour cream and remaining 1/2 teaspoon of vanilla.

Serve each slice of pie with a dollop of topping.