The skills that made you a strong engineering manager are the entry ticket here, not the whole job. What changes is scale: you plan across teams, build managers instead of engineers, and speak for engineering in rooms where nobody reads code. Learn them in this order and you'll spend less time guessing.
Managed multiple engineering teams and improved delivery.
Led 4 teams (5 managers, 38 engineers) through a platform migration, cutting deploy lead time from 3 days to 4 hours and reducing regretted attrition to zero over 12 months.
No. Hiring managers care about the teams you've run and the results they produced. A course can help you learn budgeting or org design, but it won't replace experience managing managers.
Technical enough to review a design doc, question an estimate and follow an incident as it unfolds. You don't need to be the fastest coder, but losing touch with how your systems work makes you easy to mislead.
Get a second team or a manager reporting to you, then own a plan that crosses team lines. Those two things show up in almost every director interview.
Writing. You stop fixing problems in person and start fixing them through planning docs, headcount requests, postmortem summaries and weekly updates to your VP. A clear page that says what changed, what is at risk and what you need gets things moving. A vague one gets you another meeting, and your calendar is already full of those.