Is the Hadeeth authentic that includes a Prophetic order for Muslims to learn half the religion from ‘Aa’ishah, may Allaah be pleased with her?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

The Hadeeth you are asking about is unauthentic. Quoting Ibn Hajar Imaam Ash-Shawkaani said that he does not know a chain of narrators for this Hadeeth. Al-Mizzi and Ath-Thahabi (the two very famous Hadeeth scholars) were asked about it and they stated that they did not come across it.

Allaah Knows best.