Place the shaved dark chocolate and halved cherries in separate bowls and refrigerate. Keeping these cold is essential so they don't melt the ice cream base when added later.
In a medium heavy-bottomed saucepan, combine the heavy cream and whole milk. Heat over medium-low, stirring occasionally, until the mixture reaches a gentle simmer (around 160°F). Do not let it reach a rolling boil.
While the dairy heats, whisk the egg yolks and sugar in a medium bowl for about 2 minutes until the mixture is thick, pale yellow, and smooth.
To prevent scrambling, slowly drizzle one ladle (about 1/2 cup) of the hot cream mixture into the egg yolks while whisking constantly. Repeat with one more ladle to gradually raise the temperature of the yolks.
Pour the tempered egg mixture back into the saucepan with the remaining cream. Cook over medium-low heat, stirring constantly with a heat-resistant spatula, until the custard thickens and reaches 170°F–175°F. It should be thick enough to coat the back of a spoon.
Immediately pour the custard through a fine-mesh strainer into a clean bowl to remove any potential bits of cooked egg. Cover the surface directly with plastic wrap to prevent a "skin" from forming and refrigerate until the base reaches 40°F (at least 6 hours or overnight).
In the final 2 minutes of the churning cycle, add the chilled shaved chocolate and cherries. This ensures the inclusions are evenly suspended without sinking or stalling the machine.
Transfer the soft-serve consistency ice cream into a pre-chilled, airtight container. Press parchment paper against the surface to prevent ice crystals and freeze for 4–6 hours until firm and scoopable.