Legally, you are not supposed to start working before the start date listed on your EAD (you should have your EAD with you before you start work). If the HR department of company asks you to submit proof of your employment eligibility on your first day and you don't have the EAD card then you will be in a embarrassing situation. However, my personal experience is that unless the HR people are used to dealing with a lot of international students, they don't know that they are supposed to check the EAD. They just look at the I-20 (at least that's what happened to me).

Working before getting your EAD is like working off-campus without authorisation. We all know how unlikely it is that anyone would ever get in trouble over that. So, if you don't mind taking the risk, you could just start working for the company but not report the work to your college until your EAD start date. When you report your employment info to your college just put the EAD start date as your employment start date. Again, legally, you are not supposed to start working before you get your EAD card, though.