Scoop pulp from large watermelon piece into large bowl, reaching into mouth to scoop out the flesh and leaving a 1-inch-thick shell on bottom and sides. Blend removed pulp, in batches, in blender until smooth. Pour watermelon purée through strainer into separate bowl, pressing strained solids with back of spoon to remove as much juice as possible. (You should have about 6 cups watermelon juice.) Discard any remaining watermelon pulp or reserve for another use.