For Topping: Using an electric mixer on medium speed, beat together the butter and 1 cup of sugar in a large bowl until smooth and creamy. Blend in egg and vanilla. Stir in flour, oats and 3/4 tsp. of cinnamon until thoroughly blended.
For Filling: In a separate bowl, combine the sliced apples, 3/4 cup of sugar, cornstarch, 1 1/2 tsp. of cinnamon and nutmeg. Mix well.
Evenly spoon the apple mixture into a 3 quart casserole dish. Spoon topping mixture evenly over apple filling.