Preheat the oven to 325°F (163°C).
Set the unbaked 9-inch deep-dish pie shell on a baking sheet. Keep it aside while preparing the filling. The baking sheet provides easier handling and helps contain any filling that may spill during baking.
Place the 1/2 cup unsalted butter in a small saucepan and melt it over medium heat.
Continue cooking the melted butter, stirring from time to time, until it develops a lightly browned color and a distinctly nutty fragrance.
Remove the saucepan from the heat and allow the browned butter to cool for 10 minutes.
Add the 2 1/3 cups packed brown sugar, 4 room-temperature eggs, 1/4 cup whole milk, 2 teaspoons pure vanilla extract, and 1/2 teaspoon salt to a large mixing bowl.
Whisk thoroughly until the ingredients are fully blended and the mixture becomes smooth and uniform.
While whisking continuously, gradually pour the cooled browned butter into the brown sugar and egg mixture. Continue whisking until the butter is completely incorporated.
Fold in the 1 cup of toasted, chopped pecans, stirring gently until they are evenly distributed throughout the filling.
Pour the completed pecan mixture into the prepared unbaked deep-dish pie shell.
Transfer the pie to the preheated oven and bake at 325°F (163°C) for 50–60 minutes, or until the filling appears puffed and the surface has developed a golden color.
If the pastry edges begin browning too rapidly before the filling has finished baking, loosely cover the pie with aluminum foil.
Remove the baked pie from the oven and place it on a wire cooling rack.
Allow the pie to cool completely so the filling can properly set. The filling will naturally settle somewhat during cooling.
For neater, more defined slices, refrigerate the cooled pie for at least a couple of hours before cutting and serving.