In my state, you couldn't. It depends on the requirements of the state as to licensing, so check with your local DMV. Usually you must be in current status or show at a minimum an employment authorization document to be able to drive.
This advice does not form an attorney-client relationship and is merely informative. It should not by itself be relied upon to address a legal concern.
When you receive an EAD work permit, you should be able to apply for a drivers license.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.