Place the cookies in a bowl, cover, and refrigerate. Whisk the eggs in a mixing bowl until light and fluffy, 1 to 2 minutes. Whisk in the sugar, a little at a time, and continue to whisk until completely blended, about 1 minute more. Pour in the cream and milk, whisking to blend. Add the peppermint extract and blend in well. Transfer the mixture to an ice cream maker and freeze, following the manufacturer's instructions. After the ice cream stiffens (about 2 minutes before it is done) add the chopped cookies and continue freezing until the ice cream is ready. Let stand to ripen and harden.
