Whisk flour, baking powder, baking soda, salt, and nutmeg together in a bowl. Using stand mixer fitted with paddle, beat butter and sugar on medium-high speed until pale and fluffy, about 2 minutes. Add egg and milk and beat until well combined. Reduce speed to low, add flour mixture and coconut, and mix until just incorporated.

Transfer dough to a counter and, using your floured hands, roll dough into 15-inch log, then flatten top and sides to measure 3 inches by 1 inch. Wrap log tightly in plastic wrap and refrigerate until firm, at least 45 minutes or up to 3 days.

Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350 degrees. Line 2 baking sheets with parchment paper.

Slice chilled dough into 1/4-inch-thick rectangles and space them 1 inch apart on prepared sheets. Using floured fork, make crosswise indentations in dough slices. Bake until toasty brown, 15 to 18 minutes, switching and rotating sheets halfway through baking. Let cookies cook on sheets for 10 minutes, then transfer to wire rack. Let cookies cool completely before serving.