If your ice cream maker is large enough, you can easily double the recipe to make 6 cups.
Combine the milk and cream in a medium saucepan. Add salt and 1/3 cup of sugar. Bring to a simmer. Prepare an ice water bath.
Whisk together egg yolks and remaining 1/3 cup sugar. Gradually whisk in half the milk mixture. Pour back into pan, and cook over medium heat, stirring constantly, until thick enough to coat the back of a wooden spoon, 6 to 7 minutes. Pour custard through a fine sieve into a bowl set into prepared ice water bath. Let chill, stirring occasionally. Mixture can be made ahead up to this point and refrigerated for up to 2 days.
Freeze in an ice cream maker until ice cream holds its shape when machine is turned off, add cookies and churn to combine about 1 minute. Transfer to a loaf pan, sprinkle with crushed cookies. Eat, soft from the machine, or freeze for up to 2 weeks. Soften at room temperature for 10 minutes before serving.