I can offer an answer based on personal experience, since in some ways I feel similarly to your employee.
I have a situation where in terms of my direct reporting, the tasks I am given do not match at all with my expectations about what my duties would be in the job. I have worked very hard for almost two years to be a team player on these tasks, assuming that once I "proved myself" then my job description would migrate back to what it was supposed to be based on the communication during the hiring process.
I have a great manager who is proactive and listens to me -- but at every turn he also tells me that absolutely nothing can be done about my dissatisfaction with my job duties. It cannot and will not change, even though I have repeatedly expressed the fact that I am not happy with the job duties I have been given, I feel they are not what I signed up for, and that I have done more than my fair share of the unpleasant work (and done it a way that has been praised by many folks in the company).
Now, I do not approach my manager with disrespect as you document about your employee. I like my manager and respect him, I just cannot tolerate the inability to change my job duties to reflect those that I was told during the hiring process.
It's obviously unacceptable for this person to publicly display insubordination and disrespect. I am not an HR worker, so I don't know the best way to handle that, but you should consult with HR to determine how to rectify the disrespect.
And I would largely keep the "disrespect" issue separate from the work issue. Since this employee displays good work ethic and efficiency in delivering things for other people, my suggestion is that perhaps the employee is very dissatisfied with the way this work could directly translate to career or experience opportunities.
Doing the same work for other people is not surprising. For other people, it builds goodwill and a reputation of good "corporate citizenship" which can help the employee advance. But doing these tasks for you, as if it is accepted that these tasks are supposed to be the primary job duties, could seem like it is pigeonholing the employee into implicitly accepting these items as primary duties when it reality they are not the things the employee wants at all.
Lastly, you should challenge your assumption that the work is "exactly the same" for other people. It might seem that way superficially for you, but in reality there could be pockets of more or less interesting topics. There might be technical achievements or new skills that can be learned because of minor variations in the way that others ask for the task to be completed. The employee may see those as learning opportunities meanwhile seeing the status quo tasks from their usual manager as "brain rotting" and as tasks that actively harm personal development.
Again, the biggest issue here is to correct the disrespect, because no one can be happy or get work done if that is going on.
But after that, take a long hard look at the assumptions you are making about this employee's job satisfaction. Are you challenging them? Are you providing learning opportunities? Have they communicated projects they prefer and have you arranged to give them lots of working time to focus on those?
I think the above is an often overlooked explanation, but there could also be a few others that I will describe with less detail:
The employee may not be happy with pay / benefits / perks. They may not feel like other managers control this stuff, but they may feel that the direct manager is not doing anything to help them get a raise. Especially if they are a very strong IT person as you put it. They may feel their skills are overlooked and under-utilized, and yet the manager is not working hard to ensure they are fairly compensated.
The employee may feel overworked. Even if, as you say, they are not delivering things fast enough. Did the employee just complete a big project recently, like within the last 6 months? Were they asked to do work on an especially bad project? One with bad infrastructure? Did they have to fix messes created by someone else? In the great book Peopleware there is a useful chapter on Overtime vs. Undertime. Employees who max themselves out cognitively, either because they must endure work that they dislike for a long time, or because they have to work too many hours, will always go through an extended phase of undertime, where they seem disconnected and are unable to get back that strong drive or work ethic. Even if you don't see it directly, you should ask yourself, and the employee, whether they have experienced this recently? If so, maybe something as simple as granting them an extra long vacation would fix the problem?
Long story short, there are many hidden reasons why a worker might feel this way that have nothing to do with legitimate insubordination, and as a good manager you should explore all of these options carefully before concluding anything negative about the worker.