Heat oven to 350°F. Spread coconut in ungreased shallow pan. Bake uncovered 5 to 7 minutes, stirring occasionally, until golden brown. Turn off oven. In food processor bowl with metal blade, place cooled coconut. Cover; process until ground. Set aside.
Meanwhile, in large bowl, beat butter and 2/3 cup sugar with electric mixer on medium speed about 1 minute or until smooth. Add egg yolks, rum and vanilla. On high speed, beat about 1 minute or until blended.
In medium bowl, stir together flour, baking powder and salt. Stir flour mixture into butter mixture until well blended. Cover; refrigerate 30 minutes.
Heat oven to 350°F. Shape dough into 3/4-inch balls. On ungreased nonstick (not insulated) cookie sheets, place balls about 2 inches apart. Dip bottom of small glass into 1/4 cup sugar; press on cookies to make about 1 1/2 inches in diameter.
Bake 10 to 11 minutes or just until edges begin to brown. Remove from cookie sheets to cooling rack. Cool completely, about 15 minutes.
For each sandwich cookie, spread about 1 teaspoon of the caramel apple dip on bottom of 1 cookie, spreading to edge of cookie. Top with second cookie. Gently squeeze until filling oozes out a little around the side. Roll edges of cookies in ground coconut.