Combine milk and eggs in a large saucepan and whisk. Add in sugar and vanilla extract, and cook over medium-low heat. Stir constantly with a wooden spoon until the mixture has slightly thickened and has reached 170° F.
Remove from heat and allow to cool. Stir in cream and peppermint essential oil.
Pour mixture through a fine sieve into a bowl. Add in the chocolate chunks and cover the bowl with plastic wrap to prevent skin from forming. Refrigerate for 3 hours (or up to 12 hours).
Freeze mixture in ice cream maker according to manufacturer's instructions. To finish freezing, put mixture into a dish (I used a bread pan), cover with plastic wrap, and allow to harden in the freezer at least 1 hour before serving.
Notes
You can substitute ¾ teaspoon peppermint extract in place of the essential oil if you'd like.