Beezel Breakout is a quest you can find in Coromon during the story in Hayville.
The quest is automatically given to you as you upgraded your Gauntlet with the Push Module by the Trainer Hub lady named Beth.
She mentions that a bunch of Beezel have gotten loose and are terrorizing the village. There are six in total and they are scattered all over Hayville.
Each time you find one you'll have to battle it and defeat it.
You can use the map above to see their locations. You cannot catch them, as they are trainer-owned, despite them teasing you with a Potent Potential Beezel by the well.
There is one inside the house near the Cabbage Patch in the bedroom.
Once all 6 have been defeated, head back to the Trainer Hub and speak to Beth for your reward.
- Stinky Disc