In a bowl, pour sherry over cherries. Cover with plastic wrap, and let soak overnight.
In a heavy-bottomed saucepan, bring whipping cream, half-and-half, and salt to a simmer over medium heat. Add cinnamon sticks, then remove from heat; cover and set aside for at least 30 minutes. To develop a more pronounced cinnamon flavor, let infuse longer. Taste to see if has reached desired level of cinnamon flavor, then strain.
In a bowl, make a custard by whisking together egg yolks and sugar until well blended and slightly thickened, then whisk in 1 cup of warm cream mixture. Whisking continuously (so as not to cook the eggs), slowly pour egg-sugar mixture into saucepan with remaining cream.
Prepare an ice-water bath large enough to hold a 2-quart bowl; set aside. Return pan to stove and cook custard over low heat, stirring constantly with a wooden spoon or heatproof spatula. Stir without stopping until custard thickens slightly and coats the spoon, about 8 minutes. Immediately remove pan from heat and strain custard into a 2-quart bowl. Place bowl in ice-water bath to cool completely. Cover custard with plastic wrap and refrigerate until thoroughly chilled, at least 4 hours.
Transfer cold custard into the bowl of an ice cream maker and churn according to the manufacturer’s instructions. Strain sherry-soaked cherries, reserving the liquid, and fold them into the fully churned cinnamon ice cream before putting it into the freezer to firm up.
Meanwhile, in a saucepan, cook the cherry liquid over medium heat until reduced to a syrup, about 8 to 10 minutes; allow to cool completely. Serve ice cream with a drizzle of cherry sauce.
Adapted from The Paley's Place Cookbook, by Vitaly Paley and Kimberly Paley with Robert Reynolds, copyright © 2008 (with permission from Ten Speed Press).