How you can approach this depends a lot upon how you currently interact with your team.
I have a roughly 50/50 mix of introverts and extroverts as direct reports, and encouraging some degree of social rapport has been beneficial in terms of the wider professional interactions that they need to have, for example in meetings.
I generally chat with all my team at some stage during the week - asking how their weekend was, or "Hows it going?" - when I arrive, leave, get coffee, waiting for a meeting to start and so on.
Responses vary from "Good, thanks" to a long description, but even the most introverted from time to time wants to share something, and its not always about their current work challenges.
This has set the tone for the team culture I aim to promote; we don't all have to be friends, but positive social interactions certainly help.
Even this small level of "social icebreaking" means that more difficult conversations are easier to have, and at an earlier stage, without it seeming like prying on either side.
In this case, however, I would suggest there is a serious performance issue to address. If they are falling asleep at their desk, then they are too tired to do their job effectively.
As a line manager it is well within your job description, and part of your professional obligations to the rest of the your team, to have a conversation with the team member over this issue.
The situation here is a little different to your previous question, as you have a concrete example of inappropriate behaviour (being asleep on a number of occasions), and easy to measure metrics (being at work, on time)
That said, personally I would still approach this in similar way, in a one-on-one, as the previous issue (link above):
"You seem really tired, and its impacting on your work here. Is
The difference in this case is, however, that you are having a "performance management" conversation based around negative feedback. The employee is acting inappropriately, as opposed to simply not delivering as expected, and if you don't take action you are condoning what they are doing.
As a final point, I would add that the "performance management" aspect of team leadership is probably one of the most difficult skill areas to grow, especially when dealing with underperforming staff. As with any skill, training and practice both help significantly, however its likely there will always be challenges that stretch your professional capabilities.