In a mixing bowl, beat the butter, brown sugar, white sugar, egg, and vanilla extract until well combined. Beat in flour, baking soda, salt, cinnamon, nutmeg, and rolled oats until thoroughly mixed. Stir in the butterscotch chips and coconut. Drop the dough by tablespoonfuls about 2 inches apart onto baking sheets.
Bake the cookies in the preheated oven just until the bottoms are light brown, 7 to 8 minutes. If you prefer crunchier cookies, bake 9 to 10 minutes.