When buying coconuts, be sure that shells are not cracked and there is liquid inside.
Heat oven to 400 degrees. Pierce coconuts at end marked by three small indentations. Allow liquid to drain into a bowl and set aside for thin coconut milk. Put coconuts in oven for 5 to 10 minutes. Remove from oven, crack open, and remove meat.
Peel off dark-brown skin. Grate meat very finely and heat gently in a double boiler. When coconut is hot, add 1 cup warm water. Press out thick milk by wrapping about 1 cup of grated coconut at a time in cheesecloth or a clean dish towel and twisting hard to extract as much milk as possible. Continue until liquid has been squeezed from all the coconut, reserving coconut for thin milk.
To make thin milk, pour reserved coconut liquid, plus enough warm water to make 6 cups, over reserved coconut, and press, using same squeezing procedure.