Put the cherries, 2 tablespoons sugar and the rum in a medium saucepan and cook over medium-high heat until syrupy, about 20 minutes. Remove from the heat and let cool, then refrigerate until cold, at least 2 hours.
Meanwhile, heat the milk and the remaining 1/2 cup sugar in a saucepan over medium-low heat, whisking to dissolve the sugar. Whisk the egg yolks in a small bowl. Gradually whisk about 1/2 cup of the warm milk mixture into the eggs, then whisk the egg mixture into the saucepan. Cook, stirring constantly, until the mixture coats a spoon and registers 168 degrees F on a candy thermometer, about 15 minutes. Remove from the heat; add the semisweet and unsweetened chocolate and the butter and set aside until the chocolate melts, about 5 minutes.
Transfer the chocolate mixture to a blender and puree until smooth. Pour into a large bowl and stir in the heavy cream and vanilla. Refrigerate until chilled, at least 2 hours or overnight.
Churn the chilled chocolate mixture in an ice cream maker. Strain the cherry mixture, reserving the syrup. When the ice cream is very thick, add the cherries and continue churning until incorporated. Transfer to a freezer-safe container. Drizzle in the reserved cherry syrup and swirl with a spoon. Cover and freeze until firm, at least 4 hours.