Springs off first, dome open, helper holding it. The two hinge springs carry the dome's weight. Every Big Green Egg method takes them off with the dome fully open before any band or hinge nut is loosened (BGE UK). No BGE document warns about it, but owners report a dome whipping open, and one was hit in the face when a band slipped (owner report). The dome and base are heavy ceramic (73 kg together): work with two people. A dealer can do it for you, and BGE UK offers paid servicing.
Big Green Egg calls no job technician-only, but its US warranty excludes replacements due to adjusting or aligning parts and says band and hinge alignment help is available from a dealer.
Every BGE method takes the springs off with the dome open (BGE UK); the grip is owner practice.