Set a wire-mesh sieve over a large bowl. Whisk together the egg yolks and ¼ cup of the sugar in a large bowl; set aside.
Warm the remaining ¼ cup of sugar, cream, milk and salt in a medium saucepan over medium heat. When the mixture begins to simmer, slowly drizzle it into the egg yolk mixture, whisking constantly, then scrape the warmed egg yolks back into the saucepan.
Stir the mixture constantly over medium heat with a heatproof spatula, scraping the bottom as you stir, until the mixture thickens and coats the spatula. Pour the custard through the strainer and stir in the vanilla extract. Place the bowl in an ice bath and stir occasionally, until the mixture reaches room temperature. Cover the bowl and place the mixture in the refrigerator and chill at least 2 hours, or overnight.
Churn the ice cream in your ice cream maker according to the manufacturer’s instructions. Once the ice cream is ready, layer it into your freezer-safe container with alternating layers of the chopped cookies, beginning and ending with the ice cream.