1) To be an EE, typically the minimum requirement is a B.S. degree in Electrical Engineering.
2) If a student has an undergraduate degree in Physics, one possibility is to apply for admission to a graduate program and ask the Graduate Program Director what remedial courses are required. Typically, programs will require students to take possibly 3-5 courses in the Circuits/Electronics/Signals areas to remedy the lack of background. If a student wants to obtain a second B.S. level degree, they should talk to the Undergraduate Program Director to see what courses will transfer from their Physics degree. It would realistically take the student 2-3 years to complete the BSEE degree because of the large number of required courses in Electrical Engineering.