In a large bowl, whisk together the heavy cream and matcha until combined (it's ok if it's lumpy – we'll smooth it out later).
Place the egg yolks in a medium bowl and whisk in the sugar and salt until combined. Heat the milk in a medium, heavy-bottomed saucepan until hot and steamy, swirling occasionally. Whisking constantly, slowly add the hot milk to the egg mixture. Return the pot to the stove and cook over medium-low heat, stirring constantly with a flexible heat-proof spatula, until the custard begins to "stick" to (form a film othe bottom of the pot and/or registers 170ºF on an instant read thermometer. Immediately pour the custard into the cold matcha-y cream to stop the cooking. Cover and chill until cold, 2-4 hours or overnight.
When the custard is cold, bring a small kettle of water to a boil. Fill a medium bowl partway with ice and cool water and place the mint leaves in a strainer. When the water comes to a boil, pour the water over the mint to wilt it, and immediately plunge the wilted leaves into the ice bath. Drain the mint and squeeze dry.
Place the wilted and dried mint leaves in a blender and add half of the ice cream base. Blend until very smooth, gradually adding the rest of the base. Strain the mixture through a fine-mesh strainer, pressing on the solids to extract all the good stuff. Stir in the peppermint flavor. Chill the mixture again until very cold, 1-2 hours and up to overnight.
Place a loaf pan or other vessel in the freezer to chill. When the ice cream base is cold, churn the ice cream in an ice cream maker according to the manufacturer's instructions. While the ice cream churns, place the chocolate in a small, heatproof bowl set over a small pot of steaming (not simmerinwater. Stir until just melted and let cool slightly. The chocolate should be cool but drizzleable by the time the ice cream is done churning.
Remove the loaf pan from the freezer and drizzle some of the melted chocolate over the bottom and sides. Spread 1/3 of the ice cream over the chocolate, and drizzle with more chocolate. Repeat until you've used up all the ice cream and chocolate, then freeze until, well, frozen, a few hours. When firm, scoop and serve!
Store the ice cream tightly covered and with parchment paper pressed directly onto the surface to discourage ice crystals from forming. Enjoy within a month or so.
Ice cream is simple to make, but do allow yourself several hours for chilling various stages of this recipe; most of the time is hands-off.Matcha comes in different grades, and the less expensive stuff intended for cooking and baking works fine here.Peppermint extracts can vary in strength so you may wish to add it drop by drop to taste if using a different brand.To churn the ice cream, I'm fond of my Kitchen Aid stand mixer ice cream making attachment, available here. Nutritional values are based on one of eight servings.