Whisk all the ice cream ingredients together in a large measuring cup. Cover with plastic wrap and refrigerate until chilled, about 1 hour. Pour the mixture into the ice cream maker and mix until it begins to look slushy and like soft serve, about 12 to 15 minutes. Transfer the ice cream to freezer container and cover the container tightly with a lid. Freeze until firm.
Preheat the grill to medium heat.
Toss the peaches in a bowl with the sugar, vanilla, and allspice. Put the peaches on grill and cook for 2 to 3 minutes per side.
Divide the peaches and ice cream among 4 bowls. Sprinkle with the crumbled shortbread cookies and serve immediately.