This is what I did to remove broken cable. Note bike must have been in easiest gear as no part of the cable was visible or pullable.
1. WD40 the shifter internals - just to make sure everything will move OK.
2. Get a metalwork/welding scriber tool and use it to apply force on the cable barrel in the same direction you feed the cable.
Basically you are putting on pressure the same is if it were under derailleur load.
3. You'll now be able to click the smaller shifter and it will ratchet the broken cable back one gear.
4. Apply force again on the cable barrel and shift gear again.
Do as many times as is necessary to get to the hardest gear, ie the position you need to be in to remove/install cables.
5. Remove broken cable with long nose pliers.