For the carousel cookies: Add the flour, oats, apricots, plums, sugar, baking powder and salt to the bowl of a food processor and process until a coarse meal is formed. Add the cheese and butter and pulse just to combine. Whisk the milk and egg yolk together and stream into the food processor while the machine is running. Process until the dough begins to form a ball. Roll the dough into a log on a lightly floured surface, wrap tightly in parchment paper or plastic wrap and refrigerate for 1 to 2 hours.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Unwrap the dough and cut into 1/8- to 1/4-inch-thick slices. Place the slices on the baking sheets, allowing 1/2 inch between each cookie, sprinkle lightly with turbinado sugar and bake for 10 to 12 minutes. Allow to cool completely.
To make the ice cream sandwiches: Sandwich some honey ice cream (or your favorite vanilla ice cream) between 2 carousel cookies; repeat with the remaining ice cream and cookies. Wrap the ice cream sandwiches in parchment paper and freeze until you are ready to eat.