Adjust an oven rack to the lower-middle position and heat the oven to 375 degrees. Line 2 baking sheets with parchment paper. Turn the flour, oats, coconut, baking powder and soda, salt, and cinnamon or nutmeg into a medium mixing bowl and whisk to combine.

Beat the butter in the bowl of a stand mixer with the flat-beater attachment until light and fluffy, about 2 minutes. Scrape down the bowl. Add both sugars and beat on medium speed until the sugar has dissolved and the mixture is light and aerated, pausing once to scrape down the bowl, about 3 minutes. With the mixer running on low speed, add the egg and vanilla and beat until incorporated; add the milk and mix briefly to combine. Detach the bowl from the mixer and scrape it down. Using the rubber spatula, stir the dry ingredients into the butter mixture until an evenly moistened dough forms.

Using a spring-loaded ice cream scoop just shy of 2 inches in diameter, form fifteen 2-inch balls and place them on the prepared baking sheets, spacing them 2 inches apart. Flatten the balls slightly with moistened palms. Bake 1 sheet at a time until golden brown on the bottoms and tops, 13 to 15 minutes, rotating the pan from front to back halfway through. Slide the parchment sheet onto a cooling rack and let the cookies cool for 15 minutes.