In a blender, combine water and 1 1/2 cups grated coconut and blend until a smooth puree, about 1 minute.
In a medium saucepan, whisk together the coconut milk and sugar over low heat until the sugar has completely dissolved. Bring to a boil by increasing the heat to medium. Boil for 30 seconds, then remove from heat. Whisk in the vanilla extract and coconut puree. Refrigerate until thoroughly chilled, 4 hours to overnight.
Whisk the refrigerated mixture until smooth. Pour into the ice cream machine and churn according to manufacturer's instructions. Transfer the mixture to popsicle molds. Gently tap the filled molds to remove any large air bubbles. Insert the stems or popsicle sticks and freeze until solid, 4 hours to overnight.
Line a baking sheet or cutting board that can fit in the freezer with parchment. Place in freezer. Cover a large plate with half of the coconut. Use a double boiler over boiling water or a microwave in 20 second increments to melt the chocolate until smooth. Allow to cool enough to still be smooth, but not hot.
Remove one popsicle from the mold and, working quickly, dip the popsicle in the chocolate and immediately roll in the grated coconut. Place on prepared baking sheet or cutting board in the freezer. Repeat with remaining popsicles, adding more coconut to the plate as needed. If you do not have a wide enough freezer to arrange the popsicles in a single layer, then place a sheet of parchment between the layers.