Add about 1/2 cup of brown sugar mixture to egg mixture. Whisk until well combined. Add egg mixture to remaining brown sugar mixture and return to heat. Cook over medium heat, whisking constantly, until mixture thickens and boils, about 6-8 minutes. Remove from heat; stir in vanilla and salt. Pour mixture into prepared pie crust. Cool at room temperature until pan is no longer hot. Cover with plastic wrap, placing directly on surface. Refrigerate until cold, at least 4 hours or up to overnight.