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.