Using a food processor, pulse the candy canes until broken into tiny pieces. Set aside
In a medium bowl, stir together the sweetened condensed milk, vanilla and peppermint extract. Set aside.
In the bowl of a stand mixer or a large bowl with a hand mixer, beat cream on high until stiff peaks form, about 4 minutes. With a rubber spatula, gently fold condensed milk mixture into whipped cream. Stir in the candy cane pieces.
Pour ice cream mixture into a loaf pan. Cover with plastic wrap or foil and place in the freezer until firm, about 6 hours.
Scoop into bowls and enjoy!