How do I sort a view based on when a specific field was last updated?
Such attributes are not available on fields, but they are available for nodes, and it sounds like that's all you need.

Views already provides a Sort Criteria for "Content: Updated date."

To sort by node statistics, make sure that the Statistics module is enabled (it's part of Core). That will provide additional Sort Criteria in Views. Add a Sort Criteria and filter by "Content statistics" to see three new options.

If you really need to use similar attributes on fields, you can use Rules to update extra fields based on whichever events you desire, and filter Views with those values.