Scrape the carrot and cut it in half, lengthwise, if more than 1½ inches thick. If using squash, peel, rinse and pat it dry, removing any seeds. Cut the cucumber in half, crosswise, if too long for your pot. Bury the vegetables deep in the pickling paste, cover and wait 6–8 hours for cucumbers and 10 hours to 1½ days for the carrot or squash to mature. They will mature more rapidly in hot weather. The cucumbers should be limp but still bright-colored, the carrots or squash firm but mellow.
Remove the pickles as they mature and refrigerate any that are not eaten immediately (they can be held for up to 4 days). Just before serving, rinse off any paste that clings to the pickles and slice into ¼-inch-thick half moons or other bite-size pieces. Serve at room temperature with a few drops of soy sauce.