The Army way was just to run a fresh round up the pipe until it stuck, then slap hard on the bolt-handle. Both generally come out together, you take the bum casing off the live round and you're good to go.
Good practice to clean the chamber after getting a stuck round.
WHY did the thing stick in the first place? Case separations are a warning that you are doing something wrong, so figure out what it is that you're doing wrong. MOST likely is cases too long for the chamber, which means stretched brass. WHY is is stretching so badly? Are you fireforming your brass and keeping it trimmed? Are you neck-sizing only? Are you partially case-sizing? Are you lubricating the INSIDES of the case-necks? What type of brass are you using? How old is it? Has it been properly annealed if it needed it? Fifty more questions.......
Don't let anyone scream "HEADSPACE!! HEADSPACE!!". If you are doing things right, it is not a problem.
And, if you are doing things right, separated and stuck casings should not occur.
Hope this helps.