Put butter and sugars in a bowl and mix on low speed until combined. Add egg and applesauce, mix until well blended, 2 to 3 minutes. Mix in oats, flour, baking soda, baking powder, and salt. Mix in plums and basil.
Using a 1 tablespoon scoop, drop dough onto baking sheets lined with parchment paper, spacing 2 inches apart. Bake cookies until golden and just set, 10 minutes. Let cool on sheets 5 minutes.