logo center of navigation-menu

As in topic – I’d like to get such effect as below:
<list item> <list item> <logo> <list item> <list item>

I want menu to be typical responsive with hamburger button.
Could anyone advise the best approach to get such effect? – Does anyone know any right theme that suits to such concept or should I code it on my own?

Thank you in advance!

The page I need help with: [log in to see the link]