30 ounces (1 quart plus a little extra) fresh strawberries, divided (see note above)
1 1/4 cups sugar, divided
4 tablespoons 80-proof liquor such as vodka or, preferably, Cointreau
2 cups half and half
1/2 cup corn syrup
1/2 teaspoon kosher salt, to taste
Scant drops of lemon juice, if needed
Hull and quarter 6 ounces (about 1 cup) strawberries, then slice quarters crosswise into very thin pieces. In a mixing bowl, combine strawberries with 1/2 cup sugar and alcohol and let stand in refrigerator for at least 2 hours and up to 2 days.
Hull remaining strawberries and purée at high speed in a blender until very smooth, about 30 seconds. Strain through a fine mesh strainer to filter out all seeds and fibers, then measure and reserve 1 1/2 cups purée. Extra purée, if there is any, can be put to another use.
In a clean mixing bowl, whisk together 1 1/2 cups strawberry purée with half and half, corn syrup, and remaining 3/4 cup sugar until fully combined. Add salt to taste, and, if mixture is too sweet, a few drops of lemon juice.
Chill in refrigerator or ice bath until base is very cold, at least 45°F, then 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.