In a blender, combine the coconut and 2 cups of warm water. Let stand until the coconut softens, about 1 minute.
Blend on high until creamy, 1 to 2 minutes.
Strain through a fine mesh strainer set over a large measuring cup or medium bowl, pressing on the solids; you should have 1½ cups strained coconut milk.
Discard the solids; set the coconut milk aside.
Assembling the Drink:
Blend all ingredients in a high-speed blender until smooth. Divide between 2 glasses.