You've probably heard the term managing up, and you've probably even done it, but have you ever been interviewing up? When I started at Microsoft hiring me was my boss's last checkbox before he moved on to bigger and better things. So, about 2 weeks after I got through New Employee Orientation and it was clear that I wasn't going to be thrown out, my boss put in his two week notice. 2 weeks later I was added to an interview loop for my new boss. We went through 3 or 4 loops before we settled on someone. Since then I've done that a couple of other times. I've also done similar interviews when making lateral changes inside MS talking to the potential hiring manager.
It's a very different experience from interviewing a teammate or even on IC on another team. For one thing the skill set you're looking for is different. Yes, you want your manager to have the technical chops to understand what you're saying, help you recognize potential issues, and provide pointers, but more and more, Engineering Managers have much more to do than to also be the tech lead for the team.
The role of an EM is to help the team succeed, not do the work. That's even more true for managers of managers and above, so what you really need to understand is how they'll support the team when no-one is around. And those are the things you want to look for. How have they supported their current team and direct reports? How do they approach conflict within the team and between teams? Are they focused on team health, individual growth, or team results? And do they have examples of how they do this, or are they just saying what you want to hear?
How can you find the answers to these questions? Getting them to sort a list of integers or optimizing conference room schedules won't tell you. You need a different approach here.
One of the best questions is how they define success. Is it just meeting goals, achieving business goals, or about team and individual growth? Knowing how they see success tells you what they value.
How do they stay in touch with their directs (and skip levels if that's a thing)? Do they see them as people or work units? How have they recognized and worked with those folks to make things better?
When was the last time you disagreed with/changed your boss's mind. This is a good one for any interview, but particularly for managers. It helps you understand who's side they're on. They team's, management, or the project itself? Are they looking for solutions to point problems, systemic issues, just reduced conflict?
Finally, what's their plan to make themselves replaceable, or at least, how do they think about succession and growth planning? How do they help the individuals on the team grow, and do they make it possible for them to explore new areas and expand their scope?
All of these things can give you insight into what kind of manager you're going to be getting. That's important because they say that people don't leave jobs, they leave managers. So whether you're looking at a new team or a new manager for your current team, you want to set yourself up for success.