Preheat oven to 350°. In a large bowl, cream butter, sugars and pudding mix until light and fluffy, 5-7 minutes. Add eggs; mix well. Beat in applesauce. In another bowl, whisk flour and baking soda; gradually beat into creamed mixture. Stir in oats and apple.
Drop dough by tablespoonfuls 2-in. apart onto greased baking sheets. Bake until golden brown, 15-18 minutes. Cool on pans 3 minutes. Serve warm or remove to wire racks to cool completely. Store in an airtight container. Freeze option: Freeze cookies in freezer containers, separating layers with waxed paper. To use, thaw before serving.