The flight engineer certificate isn't part of your ATP certificate. It has nothing to do with piloting, or with your piloting experience. Just like a parachute rigger certificate, a mechanic certificate, or a navigator certificate, a flight engineer is an entirely diferent certificate issued by the FAA.
Upon the flight engineer certificate, you may receive three different ratings. These are Piston, Turbopropeller, and Turbojet. There is no basic certificate without these ratings. When you take a practical test for the certificate, you do it in one of these three categories and receive the rating for that category.
The knowledge exams for flight engineer are divided into several different test gropings, and these do include a "basic" and a category rating. You need to take both. In this case, both the basic written or knowledge test, as well as the turbojet test. You take both together, and get one test result.
There are two different knowledge exams for the turbojet FE. One is FEX, and the other is FEJ. FEX contains both the basic and turbojet tests. You cannot take only the basic.
FEJ is for an add-on rating. You must already hold a flight engineer certficate. This would be taken if you hold a flight engineer certificate with either a piston or turbopropeller rating, and only want to add-on your turbojet rating.
As you have no flight engineer certificate, and you desire turbojet privileges, you must take the FEX test, and then a practical test in a turbojet aircraft/simulator requiring a flight engineer.