In a plastic bag with a rolling pin, crush candy canes into fine crumbs. It is okay to have a few larger pieces. Set candy crumbs aside.
In the bowl of a stand mixer, whip cream cheese until fluffy. Add in sweetened condensed milk, 1 tsp peppermint extract, 2/3's of the crushed candy cane crumbs and EITHER red food coloring OR 1/4 pomegranate juice. Set aside in another bowl. Save the remaining candy for the topping.
Whip 1 pint of heavy cream until fluffy and stiff peaks appear. Fold the whipped cream into the cream cheese mixture gently so as not to deflate. Fill the spring-form pan with the cream cheese mixture. Cover well with plastic wrap and freeze at least 8 hours or overnight.