Preheat your grill to medium heat, or preheat your oven to 450°F. Cut four sheets of non-stick aluminum foil, each approximately 12x18 inches in size.
In a medium bowl, combine diced apples and lemon juice. Add cornstarch, granulated sugar, brown sugar, cinnamon, and cold butter pieces. Stir well to combine. Set aside.
In a separate bowl, combine oats, flour, brown sugar, and cinnamon. Mix together. Add cold butter pieces and cut into the mixture using a pastry cutter or two forks until the mixture resembles coarse crumbles.
Place half of the apple mixture onto the center of one sheet of non-stick foil, making sure the apple mixture is touching the non-stick side. Sprinkle half of the crumb topping evenly over the apple mixture.
Fold the foil up into a tightly sealed packet, making sure all sides are securely closed. Wrap the packet a second time with another piece of foil. Repeat this process with the remaining ingredients to make a second packet.
Place packets on grill or campfire grate and cook for approximately 8–12 minutes, or place packets on a cookie sheet and bake in the oven for approximately 15 minutes. Grill and campfire times are approximate since heat levels vary.
Remove packets from heat. Allow to cool for approximately 10 minutes - the inside will be very hot. Carefully open foil packets (watch for steam). Top each serving with a scoop of vanilla ice cream and serve immediately.