Heat cherries, 1 cup heavy cream, milk, sugar, and salt in a saucepan over medium heat until mixture is steamy, 5 to 7 minutes. Reduce heat to medium-low and continue cooking for 15 minutes more.
Pour cherry mixture into a blender no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until smooth. Transfer blended mixture to a large bowl and stir in remaining 3/4 cup heavy cream.
Chill in the refrigerator until completely cold, about 2 hours; stir in amaretto liqueur, lemon juice, and cornstarch. Pour into the container of an ice cream maker and freeze according to the manufacturer's instructions.
Serving: take it out for 5-10 minutes before serving, or longer, so it comes to the right scooping temperature. Easy suggestion: ice cream is best served in chilled glass or porcelain bowls. Scoop it with a hot, but dry, ice-cream spoon (soak it in boiling water, then wipe dry).