In a medium-sized saucepan, bring the half-and-half and cream to a gentle simmer over medium heat, stirring occasionally to prevent a skin from forming on top. Remove from the heat and stir in the mint leaves. Cover and let steep for 2 hours (or overnight in the refrigerator).
Place a fine mesh strainer lined with cheesecloth over the same saucepan (a nut milk bag also works well here).
Purée the mixture in a blender, then pour through the prepared strainer. Use a spatula to press out any cream seeped into the mint; that’s where the best flavor is!
Place the saucepan over medium heat and allow the cream to once again come to a gentle simmer. Meanwhile, vigorously whisk together the yolks and sugar in a large bowl. Keep whisking until the yolks thin out slightly and become more pale in color, which means the sugar has dissolved. If you lift the spatula, a ribbon of yolks should fall back into the bowl.
Once the cream is at a gentle simmer, slowly pour it into the yolks while whisking. Turn the heat to low and return the mixture to the saucepan, scraping everything from the bowl with a spatula.
Using a spatula, stir the mixture for several minutes in a figure 8 motion, until the liquid has thickened slightly, approximately 5 minutes. If you want to use a thermometer, the temperature should read 165 degrees F.
Remove from the heat and stir in the vanilla and peppermint extracts as well as the vodka, if using. Transfer the mixture to a clean bowl and cover with plastic wrap, pressing the plastic directly against the liquid to prevent a skin from forming.
Chill thoroughly, and then prepare in your ice cream maker according to manufacturer instructions. In the final minute or two of spinning the ice cream, add in the chopped chocolate. You can also stir it in by hand.
Adding alcohol to ice cream lowers the freezing point, making it more scoopable right out of the freezer. You can’t taste any vodka, I promise! It’s not essential to the recipe, but I recommend it for a wonderful consistency.