Prepare the Ice Cream Base: Warm the milk, sugar, 1 cup of the heavy cream and the salt in a medium saucepan over low heat, stirring until the sugar is dissolved, then increase the heat to medium. Pour the remaining 1 cup heavy cream into a large bowl and place a fine-mesh sieve on top. Whisk the egg yolks in a medium bowl. Once the milk mixture begins to simmer around the edges, slowly pour it into the egg yolks, whisking constantly. Scrape the mixture back into the saucepan.
Place the saucepan over medium heat and stir constantly with a rubber spatula, being sure to scrape the bottom of the pan as you stir, until the mixture thickens and coats the back of the spatula, a few minutes. The mixture should register 170 to 175 degrees F on an instant-read digital thermometer.
Pour the custard through the fine-mesh sieve and stir it into the cream. Stir in the vanilla extract, and place the bowl over an ice bath. Stir occasionally, until the mixture is cool. Cover and transfer the custard to the refrigerator until completely chilled, at least 8 hours or overnight.
Prepare the Cookie Dough: Combine the butter and both sugars in a mixing bowl and beat on medium speed until light and fluffy, 2 to 3 minutes. Add the cream, vanilla and salt, and mix for another minute or so, until combined. Scrape the sides of the bowl, then reduce the mixer speed to low and gradually add the flour, mixing just until incorporated. Using a rubber spatula, stir in the chocolate chips. Refrigerate for at least 30 minutes, or up to 2 days.
Churn the Ice Cream: Just before churning the ice cream, scoop the cookie dough into ½-teaspoon to 1 teaspoon-sized balls and place on a plate or parchment-lined baking sheet and place in the freezer. Churn the ice cream according to the manufacturer's instructions. When finished churning, fold in the chunks of cookie dough and transfer to an airtight container. Freeze until firm, at least 4 hours.