DoE is actually a multi-functional role responsible for interfacing everything Engineering with Strategic Management, Marketing and Sales. As such Core Engineering competency is the necessity and without that everything else is useless. They must have in depth knowledge of the domain and the specific product line(s) and not mere "hand-wavy" overview. Too many DoEs err on the side of Management and not enough Engineering. Delegate appropriate responsibilities to Tech. Leads/Architects/Sr. Engineers but still be involved so that everything comes together as a organic whole. Shield Engineering from management/operational BS and politics. Have regular meetups with Product Teams and ALL individual engineers one-on-one (eg. brown bag lunches) so that you can assess motivation/morale and work on methods to maintain it at optimum level. Encourage "skunkworks" projects. Be generous in crediting individual engineers and not mere "team talk". While focusing on the company's product vision be also mindful of each engineer's aspirations and provide for opportunities for each of them to grow (eg. training in new technologies and management, 15% time on whatever side projects). Finally, always, always LISTEN to your Engineers on major tech. related decisions.
See also;
https://medium.com/connect-the-dots/what-does-a-director-of-...
https://www.quora.com/What-qualities-and-skills-separate-an-...