What was placed in the high priest's breastpiece of judgment, over Aaron's heart?

Answer: The Urim and Thummim

The Urim and Thummim were placed in the breastpiece of judgment, so they would be over Aaron's heart before the LORD.

“And place the Urim and Thummim in the breastpiece of judgment, so that they will also be over Aaron’s heart whenever he comes before the LORD. Aaron will continually carry the judgment of the sons of Israel over his heart before the LORD.”— Exodus 28:30 (BSB)