In a large bowl using an electric mixer, beat the butter, vanilla, and sugars until light and fluffy. Add the egg and beat until well combined.
Using a spoon, mix in the flour and chocolate chips, stirring until the mixture just comes together.
Place the cookie dough in a large Weber Ware Frying Pan and press down to evenly spread. Cover and refrigerate until firm, about 1 hour.
Preheat the barbecue for indirect cooking. If using a Weber Q, use a convection tray and trivet.
Once the barbecue is preheated, set the control knobs to the roast setting. Place the pan in the center of the grill, ensuring the handle is removed. Bake the cookie until golden, 12 to 14 minutes. Remove from the barbecue using the detachable handle. Allow to cool for at least 5 minutes on a cooling rack. Serve with vanilla ice cream.