There is an Upanishad named Vajra-sūcika, which contains details about the nature of Brahmin Varna. It discusses the qualities of a Brahmin. I came to know about the existence of this Upanishad through this answer.
There is no universal agreement on the authorship of the scripture. The author can be either Shankaracharya or Aśvaghosa.
The Upanishad is often ascribed to the Brahmin Buddhist scholar Aśvaghosa, the celebrated author of the 'Buddha-charita' — a biography on Lord Buddha. Sometimes, it is also ascribed to Bhagavatpāda Shankaracharya, the great Advaitin teacher. These ascriptions are impossible to prove.
There are many writings about Adi Shankaracharya by some elite people in the lineage of Adi Shankaracharya. Does anyone attribute this scripture to the Adi Shankaracharya in their writings?