Hello, Brian.
Ok, I've forgotten about Status field and we're talking only about
"Requestor.EmailAddress = tom OR Owner = tom". I think that I've found
problem. Here are three queries I want you to compare. Other user can
do that too, but please replace "main.Owner = ####" with some valid
user ID as it's very important for mysql optimizer.

The first one is the query we build in 3.6.4:
SELECT SQL_NO_CACHE COUNT(DISTINCT main.id)
FROM Tickets main
   CROSS JOIN Users Users_3
   JOIN Groups Groups_1
       ON Groups_1.Domain = 'RT::Ticket-Role'
       AND Groups_1.Type = 'Requestor'
       AND Groups_1.Instance = main.id
   LEFT JOIN CachedGroupMembers CachedGroupMembers_2
       ON CachedGroupMembers_2.MemberId = Users_3.id
       AND CachedGroupMembers_2.GroupId = Groups_1.id
WHERE
   main.Status != 'deleted'
   AND ( main.Owner = '58936'
       OR ( Users_3.EmailAddress LIKE '%tom%'
            AND CachedGroupMembers_2.id IS NOT NULL )
   )
   AND main.Type = 'ticket'
   AND main.EffectiveId = main.id;

This is one way to fix it:
SELECT SQL_NO_CACHE COUNT(DISTINCT main.id)
FROM Tickets main
   CROSS JOIN Users Users_3
   JOIN Groups Groups_1
       ON Groups_1.Domain = 'RT::Ticket-Role'
       AND Groups_1.Type = 'Requestor'
       AND Groups_1.Instance = main.id
   LEFT JOIN CachedGroupMembers CachedGroupMembers_2
       ON CachedGroupMembers_2.MemberId = Users_3.id
       AND CachedGroupMembers_2.GroupId = Groups_1.id
WHERE
   main.Status != 'deleted'
   AND ( main.Owner = '58936' OR CachedGroupMembers_2.id IS NOT NULL )
   AND Users_3.EmailAddress LIKE '%tom%'
   AND main.Type = 'ticket'
   AND main.EffectiveId = main.id;

Another way to fix the problem:
SELECT SQL_NO_CACHE COUNT(DISTINCT main.id)
FROM Tickets main
   JOIN Groups Groups_1
       ON  Groups_1.Domain = 'RT::Ticket-Role'
       AND Groups_1.Type = 'Requestor'
       AND Groups_1.Instance = main.id
   LEFT JOIN CachedGroupMembers CachedGroupMembers_2
       ON CachedGroupMembers_2.GroupId = Groups_1.id
   LEFT JOIN Users Users_3
       ON CachedGroupMembers_2.MemberId = Users_3.id
WHERE
   main.Status != 'deleted'
   AND main.Type = 'ticket'
   AND main.EffectiveId = main.id
   AND ( main.Owner = '58936' OR Users_3.EmailAddress LIKE '%tom%' );
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