Make the coconut filling: In a medium heatproof bowl set over a pan of simmering water, whisk together the confectioners' sugar, butter, egg white, and salt until very liquid and warm to the touch, about 10 minutes.
Remove from the heat and, with a spoon, stir in the coconut and vanilla until well combined. Cover with plastic wrap and refrigerate for 1 to 2 hours, up to overnight.
When ready to coat the coconut, in a small heatproof bowl set over a pan of simmering water melt the chocolate. When almost melted, turn off the heat and let the chocolate continue to melt completely, stirring occasionally. Keep the bowl over the warm water.
Form the balls: Line a small tray with parchment paper. Using a scant ounce of the coconut mixture, roll into a small ball and place on the parchment-lined tray. Repeat with the remaining mixture, forming 18 balls.
Arrange 18 paper or foil minicups on the tray. Place additional coconut in a shallow pan or bowl. Gently place 1 of the balls into the warm melted chocolate and, using 2 forks, roll the ball in the chocolate until well coated. Lift (do not pierce) the coated coconut ball with one of the forks, allowing some of the chocolate to drip back into the bowl. Roll in the additional grated coconut, if desired, and carefully place in 1 of the prepared cups. Repeat with the remaining coconut balls and melted chocolate. Refrigerate until the chocolate has hardened and use as desired.
*If using milk chocolate, add 1 teaspoon of vegetable oil while melting.