Hi, I'm just starting a farm in PNW and have asked the same question of some of the established farmers in my area. Some have indeed used pigs, successfully, to deal with scotch broom infestations!
As has been said already, the pigs are smart enough to know not to eat the broom.
Get a rooting breed, not a grazing/browsing breed as you want them digging up the roots. Use grains around the base of the broom plants to direct the pigs' rooting.
Best time to attack broom is when it's in full bloom (that would have been May this year). The plant has moved much of it's energy from it's roots to it's blossoms. After that point it is making seeds (undesired) and photosynthesizing to start putting energy back into it's roots for next year (also undesired). You may want to chop them down before much of the regenerative part of the cycle happens.
As for using goats to nip broom sprouts, I've also heard from the local farmers that goats will eat the broom, but only if there isn't much else to chew on so I'm not sure if it's all that good for them to eat. More research required, I think.
Also as was mentioned, scotch broom is a pioneer plant trying to improve poor/damaged soil. After a good rooting/chopping, might be good idea to sow some clover (or your preferred substitute) for nitrogen-fixing, ground-covering goodness to reduce both the footprint for the broom to grow back in and the need for it to grow back.
Please keep in mind - none of this is first hand experience for me, just information I'm getting from the experienced locals. If you try any of this out, I would love to hear how it goes. If I get to if first, I will let you know how it goes!