It’ll be ugly as all getout from the inside, but this is probably the only thing you can do aside from replacing the drawer or replacing the backer board.
That said, you should probably fix the primary reason this broke. That much force means:
the drawer guides may also need work or it will happen again.
the items in the drawer which may have jammed it shut should get reorganized to keep it from happening again.
the temper may need some work if this was purely a result of unloading frustration on furniture. This fix may not cost a significant amount, but a bad temper can generate drastically more financial (and sometimes criminal) damage than moderate self control. I know this one well.
Liberal application of wood glue to all mating surfaces. Then clamp in place and let cure.
The cured wood glue will be stronger than the particle board the drawer is made from.
It’ll be ugly as all getout from the inside, but this is probably the only thing you can do aside from replacing the drawer or replacing the backer board.
That said, you should probably fix the primary reason this broke. That much force means:
Or kids who have trouble remembering that kitchen cabinets and drawers are actually not indoor climbing walls
True. Hard to fix that one though.
I’d probably also add some Brad nails or staples so that it’s not just glue.
Second the wood glue. It’s stronger than wood