Can an HOA board member give their proxy to someone else?

The law provides that if authorized in the bylaws, a board member may grant a directed proxy to someone else who may or may not be a board member.  See Utah Code Subsection 16-6a-816(4).  A directed proxy is a written proxy authorizing someone to cast a vote in a way that is specified in the proxy. In other words, a directed proxy directs the person to cast the director's vote in a specified way.  For instance, a directed proxy might state, "At the meeting on June 1, 2022, I direct my proxy to cast my vote in favor of hiring XYZ Landscaping Co., pursuant to the bid submitted by XYZ."

The director is considered present at a meeting for quorum purposes if a directed proxy is used, but, again, it must be authorized in the bylaws, which in practice is quite rare.