Heat the oven to 375°F. Line two baking sheets with foil.
Stir together the flour, baking powder, baking soda, salt, and cinnamon; set the mixture aside.
In a medium bowl, beat the butter and granulated sugar with an electric mixer on medium speed until light and fluffy, 2 minutes. One at a time, add the brown sugar, egg, applesauce, and vanilla extract, beating thoroughly between each addition.
One at a time, gently stir in the flour mixture, oats, and raisins.
Drop the dough by teaspoonfuls about 2 inches apart on the prepared baking sheets. Use a fork to gently flatten the mounds of dough.
Bake until the cookies are lightly browned and look dull on the surface but are still soft and moist, 8 to 10 minutes.
Grab and slide each foil sheet of cookies from the pan to a rack to cool for 10 minutes. Remove the cookies from the foil to cool completely.
Tip: To retain the chewy texture of these cookies, avoid overbaking them. They may look slightly underdone but will reach just the right firmness when cooled. If you prefer crispy oatmeal cookies, bake them until the tops are well browned and look nearly dry.