1. Line a large colander or sieve with 5 layers of dampened cheesecloth, allowing the cheesecloth to extend over outside edges of colander; place colander in a large bowl.
2. Heat buttermilk in a large, heavy saucepan over medium-high heat. Attach a candy thermometer to edge of pan so that thermometer extends at least 2 inches into buttermilk. Cook until candy thermometer registers 170° (about 20 minutes), gently stirring occasionally. Stir in juice. As soon as buttermilk mixture reaches 170° again, stop stirring (whey and curds will begin separating at this point). Continue to cook, without stirring, until thermometer registers 190°. (Do not stir, or curds that have formed will break apart.) Immediately remove pan from heat. (Bottom of pan may be slightly scorched.) Using a slotted spoon, gently spoon curds into cheesecloth-lined colander; discard whey, or reserve for another use. Return colander to bowl; cover with plastic wrap. Refrigerate 8 hours or overnight.
3. Scrape cheese into a bowl. Add chives, cilantro, parsley, salt, and freshly ground black pepper; toss gently with a fork to combine.