Mix the brown sugar, half-and-half, butter and salt in a saucepan over medium-low heat. Cook while whisking gently for 5 to 7 minutes, until it gets thicker. Add the vanilla and cook another minute to thicken further. Since the half-and-half and butter are lower in fat, it might take a few more minutes to thicken than it would with regular ingredients.
Turn off the heat, cool slightly and pour the sauce into a jar. Refrigerate until cold.