To make the butterscotch, put 6 tablespoons of the butter and the corn syrup in a heavy saucepan. Cook over medium heat for 5 minutes, stirring occasionally. Then add the brown sugar, granulated sugar, salt, and remaining 2 tablespoons butter. Continue cooking, while whisking, for another 5 minutes. Let cool, then store, covered, in the refrigerator until you need it for the shake—for up to 4 days.
For each shake, put 2 scoops of ice cream and 2 tablespoons each of the prepared butterscotch and the half-and-half in a blender. Puree until smooth. Pour into a cold tall glass and top with whipped cream.