In a large bowl, beat butter and sugars with a mixer at medium speed until fluffy, stopping to scrape sides of bowl. Add egg; beat well. In a medium bowl, whisk together flour, baking soda, cinnamon, salt, ginger, nutmeg, and allspice. Add flour mixture, pineapple, banana, toasted pecans, and vanilla to butter mixture, beating just until combined. Cover and refrigerate until firm, about 2 hours.

Preheat oven to 350°. Line baking sheets with parchment paper.

Shape dough into 1-inch balls. Roll each ball in coconut. Place 2 inches apart on prepared pans.

Bake until lightly browned around edges, about 17 minutes. Let cool on pans 3 minutes. Remove from pans, and let cool completely on wire racks.

In a small bowl, whisk together confectioners’ sugar and 2 tablespoons reserved pineapple juice. Drizzle over cookies.