Quarter 6 ounces (about 1 cup) strawberries, then slice quarters crosswise into very thin pieces. In a mixing bowl, combine strawberries with ½ cup sugar and alcohol and let stand in refrigerator for at least 2 hours and up to 2 days.

In a countertop blender, process remaining strawberries until very smooth, about 30 seconds. Strain through a fine-mesh strainer to filter out all seeds and fibers, then measure and reserve 1 ½ cups purée. Extra purée, if there is any, can be put to another use.

In a clean mixing bowl, whisk together 1 ½ cups strawberry purée with half-and-half, corn syrup, and remaining ¾ cup sugar until fully combined. Add salt to taste, and, if mixture is too sweet, a few drops of lemon juice.

Chill in refrigerator until base is very cold, at least 45°F (7°C), about to 2 to 3 hours. Churn in ice cream maker according to manufacturer's instructions. In the last minute of churning, retrieve strawberry mix-ins from the refrigerator, strain off syrup, and add mix-ins to the churn; reserve strawberry syrup for another use (it makes a great daiquiri). Transfer ice cream to airtight container and chill in freezer for at least 4 hours before serving.