Prepare an ice bath in a large bowl and set a medium bowl in the ice bath. Set aside.
In a large, heavy bottomed saucepan over medium heat, whisk together 2 cups heavy cream, 2 cups whole milk, 1/2 cup granulated sugar, and 1 teaspoon vanilla bean powder. Continue to heat, stirring occasionally, until the mixture is just below a simmer.
Meanwhile, whisk the remaining 6 tablespoons granulated sugar and 12 egg yolks in a large bowl until slightly thickened and the whisk leaves a trail. Slowly, while whisking, add about 1/2 cup of the hot milk mixture to the yolk mixture, then whisk in the remaining milk mixture. Set a fine-mesh sieve over the original saucepan and strain the liquid back into the pan.
Cook the mixture over medium heat, stirring constantly and scraping the bottom and sides of the pan with a wooden spoon, until steam begins to rise from the surface and the custard thickens enough to coat the back of a spoon. Remove pan from heat. Place the fine mesh sieve over the medium bowl in the ice bath and strain the mixture into the medium bowl. Whisk in 1/4 teaspoon kosher salt. Let the mixture cool in the ice bath for about 30 minutes, stirring from time to time. Once the mixture is cooled, press plastic wrap against the surface of the custard to prevent a skin from forming, and transfer to the refrigerator to chill until cold, but preferably overnight. The colder the mixture is, the less time it will take to freeze in the ice cream machine, and the better the texture will be.
Pour the custard into an ice cream maker and freeze according to the manufacturer's instructions. When the ice cream is thick and creamy and almost finished (it should have the texture and look of soft serve), microwave 1/4 cup dulce de leche on high in short bursts of 10 to 15 seconds until soft and pourable. Pour the dulce de leche slowly through the opening in the top of the ice cream machine and allow it to solidify and break up in the ice cream for about 2 minutes; the dulce de leche will solidify in small chunks throughout the ice cream. Drop in the cookie dough nuggets and allow the ice cream machine to stir them throughout the ice cream.
Pack the ice cream into a storage container, pressing a sheet of parchment directly against the surface and sealing with an airtight lid. Freeze in the coldest part of the freezer until firm, at least 4 hours.