Arrange oven rack in middle position and top with heavy baking sheet. Preheat oven to 425°F.
On lightly floured surface using lightly floured rolling pin, roll out dough to 13-inch round. Fit into 9-inch pie plate. Trim edge, leaving 1/2-inch overhang, then fold over and crimp decoratively. Chill shell 30 minutes.
Scatter walnuts in shell. In medium bowl, whisk together eggs, maple syrup, butter, vanilla, brown sugar, and salt.
Pour mixture into pie shell over nuts.
Transfer pie to baking sheet and bake 10 minutes. Reduce oven temperature to 350° F and bake until crust is golden and center is just set, 25 to 30 minutes more.
Transfer pie to rack and cool at least 10 minutes before cutting.
