Fresh sour cherries have a very short season and can be hard to find, so I've created this recipe with canned sour cherries. If you have fresh available, you can certainly use those instead. You can also swap in sweet cherries, but if you do that, don't add any additional sugar to the cherries.
27ozcan sour cherries( I used Oregon Specialty Fruit Red Tart Cherries, you can substitute fresh cherries if available)
To make the ice cream:
In a medium saucepan, whisk together the eggs, sugar, lemon juice, lemon zest, milk, and cream. Place the pan over medium heat and continue to whisk regularly as it comes to a simmer. Heat and whisk until it thickens and coats the back of a spoon. If you want to use a thermometer, you're looking for it to reach 170-180 F on the thermometer.
Pour the ice cream custard through a strainer into a bowl or container, to remove all the lemon zest and any pieces of cooked egg. Add the lemon extract and whisk it in. Cover and chill the custard until completely cold.
While the custard is chilling, prepare the cherries. Place the cherries and their juice, the sugar, and the cinnamon in a saucepan, and bring to a simmer over medium heat. Simmer for 15-20 minutes, until it is bubbly and a bit thicker. (The cherry juice won't reduce by too much.) Cool to room temperature, then strain the cherries out and chop them coarsely.
Churn the lemon ice cream in your ice cream maker according to the manufacturer's directions. When the ice cream is done churning, add the chopped cherries and stir them in briefly just to get them well-distributed. Chill the ice cream for 30-45 minutes until firm enough to scoop.