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Brush the apple slices with the vegetable oil. Lay the apples on the hot grill, and turn the slices over after achieving golden grill marks. Transfer the apple pieces to a baking dish.
Melt the butter, brown sugar and 2 teaspoons cinnamon in a pan over medium heat, stirring until the brown sugar has dissolved. Remove the pan from the heat. Pour the glaze over the apples and bake until the slices have softened, approximately 10 minutes.