In a medium bowl, toss the cherries with the liqueur. Cover and let soak for at least 8 hours or overnight. Strain, reserving the liquid and cherries separately.
In a small saucepan, combine 3/4 cup water and 3/4 cup sugar. Cook, stirring, over medium-high heat until the sugar is dissolved. Transfer to a large bowl and let cool. Whisk in the cream and milk.
Using an ice cream maker, process the milk mixture according to the manufacturers instructions. In a small saucepan, bring the reserved cherry juice and the remaining 2 tbsp. sugar to a boil, whisking. Cook over high heat until syrupy, about 5 minutes; remove from the heat and refrigerate. About 5 minutes before the ice cream is finished churning, mix in the reserved cherries and the chocolate. To serve, drizzle with the cherry syrup.