Why not just have two bodies like every other parliamentary system? Europeans elect an MP and each member state elects a Senator. Then the majority party in the Parliament appoints a Prime Minister, who can appoint a cabinet. Nobody is more than two steps away from an elected position.
That makes perfect sense indeed, although personally I'd prefer two senators per state. But since the EU as a political union is an afterthought of an economic union, it's full of loose ends like this. Also, national governments will be very reluctant to give up this kind of power.
