Is it permissible for a veiled female doctor to cover her entire face, except for her eyes, in order to see clearly, and should she cover her entire face outside of work?

There is no objection to wearing a niqab that only shows the eyes, to enable the woman to see. This is mentioned in Al-Qurtubi's interpretation of the verse: "They should draw their outer garments over themselves" (Qur'an 33:59), where Ibn Abbas and Qatadah said: "That is, she wraps it above the forehead and ties it, then turns it over the nose, even if her eyes appear, but it covers the chest and most of the face."

However, one must be careful not to reveal the eyes in a manner that invites temptation, nor to wear a niqab that shows large parts of the eye sockets and cheeks, as this constitutes displaying adornment.