Set up grill for indirect cooking over medium heat (no heat source directly under cookies).
Soften ice cream on the counter for 10 minutes. Scoop large balls of ice cream onto a parchment-lined baking sheet, spacing about 3 inches apart. Top the scoops with another piece of parchment. Gently flatten the scoops into 2-inch disks about 1-inch thick. Put the baking sheet in the freezer to harden while making the cookies.
In a mixing bowl, combine rest of the ingredients. Stir to form dough. Drop by rounded tablespoons onto foil baking sheet. Place foil baking sheet on another foil baking sheet (for stability and insulation) and set on grill away from heat. Cover grill and cook 12 to 16 minutes or until just baked and browned.
Remove from grill and let cool to room temperature.
Sandwich the ice cream disk between to cookies and wrap in parchment or waxed paper until ready to serve.
INDOOR: Preheat oven to 375 degrees F. Prepare cookies as directed and drop onto an ungreased cookie sheet. Bake in preheated oven for 10 to 14 minutes. Cool for 1 minute on cookie sheet before transferring to wire rack. Cool cookie completely, and then freeze for a minimum of 1 hour before completing ice cream sandwiches as directed.