Software engineering absolutely. You need to be self-driven and somewhat ambitious but many (myself included) have done it. EE I'm not sure.
In my department, not without an MS, though there is one guy working on it part time. You also need to be self driven and ambitious.
There seems to be a glut now of pHD, probably due to the shitty economy over the last several years. Might as well just stay in school.