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.