In a saucepan, combine the milk, heavy cream, and sugar. Heat over medium heat, stirring until the sugar is completely dissolved, but do not boil.
Add the fresh mint leaves and simmer over low heat for 10 minutes to infuse the mixture with mint flavor.
Remove the saucepan from the heat and strain the mixture to remove the mint leaves.
Stir in the finely chopped dark chocolate and vanilla extract, then add the food coloring if desired for a more vibrant green color.
Allow the mixture to cool completely at room temperature, then refrigerate for at least 2 hours to chill thoroughly.
Pour the chilled ice cream base into an ice cream maker and churn according to the manufacturer's instructions. If you don't have an ice cream maker, pour the mixture into a freezer-safe container and stir with a fork every hour to prevent ice crystals from forming.
Place the ice cream in the freezer for at least 4 hours to harden completely.
Before serving, let the ice cream stand at room temperature for a few minutes to soften slightly for easier scooping. Garnish with extra chocolate shavings and fresh mint leaves, if desired.
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