In a medium bowl, mix together the dark brown sugar and flour, breaking up any sugar lumps. Add 1 cup of whole milk and the egg yolks. Whisk until smooth and set aside.
In a 4-quart saucepan, melt the sugar over medium-low heat. Shake the pan periodically to evenly heat the sugar. Once the edges start to melt, stay with the saucepan and swirl until all the sugar is melted. Continue to cook until the sugar reaches a deep amber color, 3-5 minutes. Remove from heat and slowly add the heavy cream and remaining ½ cup of whole milk. The mixture will sputter and the caramelized sugar will harden.
Return the mixture to medium-low heat and stir until the caramel is melted into the milk mixture, 3-5 minutes. Slowly whisk in the prepared egg yolk mixture. Continually whisk and cook over medium heat until the mixture thickens. Keep whisking until it's just beginning to bubble throughout and thick, 5 minutes.
Remove from the heat and add the salt, vanilla extract and butter. Whisk until the butter is melted and the pudding is smooth. Pour into the prepared pie shell. Press plastic wrap directly on the pudding surface to prevent a skin from forming. Place in the refrigerator and chill until firm, at least 6 hours.
For the meringue
Bring a saucepan with a few inches of water to a slow simmer on the stove over medium-low heat.
In a large bowl (I use the bowl of my electric mixer), combine the egg whites and sugar. Whisk to combine. Set over the slightly simmering water and whisk until the sugar is dissolved or the mixture reaches 160°F, 5 minutes.
Remove from the heat and add the salt and cream of tartar. Using an electric mixer, whisk until stiff peaks form, 5-7 minutes. Whisk in the vanilla extract.
Spread meringue on pie creating peaks. Place the pie under a low broil. Watch closely and brown the meringue until golden, 1-2 minutes. Remove and serve immediately.