Heat oven to 375° F. Line 2 baking sheets with parchment paper or aluminum foil.
With an electric mixer on medium-high, beat the butter, brown and granulated sugars, corn syrup, and vanilla for 3 minutes. Add the egg and beat until combined. In a separate bowl, whisk together the flour, salt, and baking soda. Reduce mixer speed to low and slowly add the flour mixture to the egg mixture.
Form the dough into teaspoon-size balls. Spread the coconut on a plate. Roll each ball in the coconut, pressing so it adheres. Place on prepared baking sheets. Bake for 10 to 12 minutes. Cool on the baking sheets for 5 minutes. Transfer the cookies to wire racks.
Meanwhile, in a heatproof bowl placed over, but not touching, simmering water, melt the chocolate. Turn half the cookies upside down and spread the flat sides with the white chocolate. Sandwich them with the remaining cookies.