You might find something usable in the faucet adapter/aerator section of your favorite plumbing supply joint. That's not a standard
setup, so it doesn't have a standard name, except "adapter". When you find one, leave off the thread tape, because that's probably
why it split. That connection almost certainly was meant to be sealed with a rubber washer or o-ring, requiring only to be made a bit
more than finger-tight.