In a heavy-bottomed saucepan, combine 1 cup dark brown sugar, ½ cup water, and ½ teaspoon salt. Bring to a boil over medium heat and cook for 5-6 minutes, until the sugar dissolves and the mixture looks thick and syrupy.
Slowly add 3 cups heavy cream and 1 cup whole milk. Be careful, this step will splatter. Stir until smooth and bring back to a gentle simmer.
In a medium bowl, whisk together 1 whole egg, 3 egg yolks, and 2 tablespoons cornstarch until smooth.
Temper the eggs by slowly adding about 1 ½ cups of the hot cream mixture, a little at a time, whisking constantly so the eggs don’t curdle.
Pour the tempered egg mixture back into the saucepan with the rest of the cream mixture. Cook over medium heat, stirring constantly, until the pudding thickens and just begins to boil.
Remove from the heat and stir in 4 tablespoons unsalted butter and 1 tablespoon vanilla extract until fully melted and glossy.
Strain the pudding through a fine-mesh sieve into a large bowl or measuring cup, then divide evenly into six 8-ounce cups or ramekins.
Press plastic wrap directly onto the surface of each custard to prevent a skin from forming. Let them cool at room temperature for about 1 hour, then refrigerate until completely set, about 3-4 hours or overnight for the best texture.