Now I have a question: If I file another I-765: Replacement EAD, will I get the same starting date as my approved but lost EAD?
-It seems you can only file an I-765 to get the replacement and check "Replacement" option in the form. But you may need to pay the filing fee for the replacement if your EAD is I-485 based [(c)(9)] even if your I-485 filing fee was new fee of $1010 and not need EAD filing fee for renewal/new application (see I-765 Instructions Page 7).
You can continue your work legally even if your current EAD expires if your employer has no probelm about that. The replacement EAD will be the same as the lost one.