Mash cooked sweet potatoes: Preheat oven to 350°F. Bring a large pot of water to a boil over high. Add sweet potatoes, and cook until tender when pierced with a fork, about 10 minutes. Drain sweet potatoes, and transfer to a large bowl; mash using a potato masher until mostly smooth. Let cool 15 minutes.
Add milk, sugars, and spices: Add evaporated milk, granulated sugar, vanilla, cinnamon, egg, ¾ cup of the brown sugar, and ¼ teaspoon of the salt to sweet potato mixture; whisk until well blended. Lightly grease a 13- x 9-inch baking dish with butter; pour sweet potato mixture into dish.
Make pecan topping: Place pecans, butter, and remaining ½ cup brown sugar and ½ teaspoon salt in a medium bowl; mix using hands until mixture resembles a coarse streusel. Sprinkle evenly over sweet potato mixture.
Bake casserole: Cover with aluminum foil, and bake in preheated oven until lightly browned, about 20 minutes. Remove foil, and bake until set, about 15 minutes.
Serve: Let cool 10 minutes. Serve.
