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Transfer the chimichurri butter to a sheet of plastic wrap and shape it into a log. Wrap in plastic wrap and refrigerate until it has hardened.
Meanwhile, bring large pot of salted water to a boil. Add the corn and boil until it is tender, about 5 to 7 minutes. Drain and transfer to a serving platter. Place a disk of chimichurri butter on each ear of corn and serve sprinkled with Cotija cheese and a lime wedge on the side.