Off the top of my head, these are the kinds of things engineering managers often have to deal with:

Hiring:

- Writing job specs.

- Dealing with recruiters

- Reading CVs/résumés

- Interviews

- Onboarding

- Outreach

- Who do you need to hire next quarter to avoid capacity problems?

People:

- How do you level up your developers? What new responsibilities can you delegate to them so that they can grow without swamping them? Are you available enough to them so you can help them take these new tasks on?

- Evaluating training courses / conferences etc.

- Are there any personal or interpersonal problems that need sorting?

- What can you do to help your team gel better and feel like they are part of a team?

- Where are your team members' careers heading and how can you help them get there?

- Your team has grown too large to manage effectively by yourself. How do you split things up?

Process:

- What is everybody working on? What will everybody work on next?

- What are people blocked on? How do you unblock them? What is likely to block them next?

- Can you do things in a better way? How?

- You've got a lot of bugs in the backlog. Is there a root cause? Are there any patterns?

- You aren't performing as well as you thought you were. Where is the time being spent? What's the cause? Are the estimates wrong or the performance?

- Some people want to shift their hours or work remotely. Can you accommodate this? Do you have to adjust process and if so, how?

Line management:

- Approving invoices

- Approving holidays

- One on ones

- Salary review

- Getting people back to work after sickness / parental leave / sabbaticals

- Disciplinaries / performance problems

- Firing / redundancies

- Exit interviews

Planning:

- How well is your team performing? How do you measure this and how do you improve?

- Do you have enough capacity?

- If not, which features do you bump?

- Are there any bottlenecks in the pipeline?

- What are you telling the shareholders you're delivering next quarter?

Product:

- Does marketing know what you are building next? What information do they need from you to sell it?

- Do CS know how to support your customers with the new features?

- Feedback from the rest of the business about the product and how the tech team works.

- Are the specs precise, correct, complete, and achievable?

- Features have been requested. Are they technically feasible? What's the general size of it and what quarter can we deliver it by?

- A big customer has a major problem with your product. It's not your fault, but it has a disproportionate affect on the customer. How do you prioritise solving this problem?

External:

- A shareholder owns a business with a related product that they want integrated ASAP. How do you deal with that pressure without disrupting your plan?

- One of your suppliers has a data breach that has leaked your data. How do you deal with that? Have you defined processes for reporting security issues?

- A supplier is failing to perform adequately. Can it be fixed? How do you move away? How soon? Where does the work fit into the schedule?

- A supplier has changed their pricing structure. Do you have to move? How soon? Where does the work fit into the schedule?

- New legislation has been passed and you have to make changes to your product. What are the requirements?

- A complaint has been made about the accessibility of your product. Does it have merit? What's the impact of fixing it, and how urgent is it?

- A big prospective customer is on the verge of signing, but they must have one feature that you weren't planning on building until next year. How do you rearrange things to get the customer with minimum impact?

- A service you use is changing their API. What's the risk of staying on the deprecated version until it can be planned in?