Against the individual: assault, battery, intentional infliction. Against the operator: negligent hiring, negligent retention, negligent supervision, negligent training, failure to report, and in states with a statutory framework, abuse or neglect under the statute with its enhanced remedies.
The operator counts are pled as negligence deliberately. Intentional acts are excluded by every liability policy, so a claim framed purely as intentional conduct reaches an uninsured defendant. Framed as negligent supervision, it reaches the operator and the operator insurance, which is where recovery is available.
Expect the pleading to reach for punitive damages, and expect prior complaints to be the centre of the case. A documented prior allegation about the same employee, or a pattern of similar complaints handled without escalation, converts a single incident into a corporate case.
Resident-on-resident aggression is pled differently and sits in a gap worth naming: many abuse endorsements define abuse as an act by an employee, volunteer or contractor, so a resident is not covered, while a broadly worded assault and battery exclusion may still catch the injury. The claim can fall between the two.