Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch pie dish.

Place the brown sugar and butter in a mixing bowl, and beat them together with an electric mixer until creamy and very well combined, without lumps. Beat in eggs, one at a time, incorporating the first egg before adding the next one. Add the vanilla extract and salt; beat the flour in, a little at a time, and then the milk, making a creamy batter. Pour the batter into the prepared pie dish.

Bake in the preheated oven for 35 minutes; remove pie, and cover the rim with aluminum foil to prevent burning. Return to oven, and bake until the middle sets and the top forms a crusty layer, about 15 more minutes. Let the pie cool to room temperature, then refrigerate for at least 1 hour before serving.