In medium bowl, beat ¾ cup butter and the granulated sugar with electric mixer on medium speed 2 minutes or until smooth and creamy.
Beat in egg and ¼ teaspoon rum extract until well blended.
On low speed, beat in flour, baking soda, cream of tartar and salt about 1 minute or until dough forms.
Shape dough into 36 (1¼-inch) balls.
Place 1 ball in each muffin cup.
Bake 10 to 12 minutes or until set and edges are light golden brown.
Using end of wooden spoon handle, press 1-inch-deep indentation in center of each cookie to form a cup.
Run knife around edges of cookie cups to loosen.
Cool completely, about 15 minutes.
Remove from pans to cooling racks.
In large bowl, beat powdered sugar and ⅓ cup butter with electric mixer on low speed until well blended.
Add ⅛ teaspoon rum extract and 1 tablespoon milk; beat until smooth and spreadable, adding additional milk, 1 teaspoon at a time, if needed for desired consistency.
Stir in pineapple.
Spoon about 2 teaspoons filling into each cookie cup.
Sprinkle with coconut.
Store on plate in refrigerator, loosely covered with plastic wrap.
Notes / Tips / Wine Advice:
Make-Ahead Magic
Make and bake the cookie cups as directed. Place cooled unfilled cups in plastic storage container or resealable freezer plastic bag; keep frozen up to 2 months. Thaw at room temperature about 30 minutes or until completely thawed. Fill as directed.
Kitchen Secrets
To toast coconut, heat oven to 350°F. Spread coconut in ungreased shallow pan. Bake uncovered 5 to 7 minutes, stirring occasionally, until golden brown.