Do I have a case if a waitress dropped hot skillet in my lap which resulted in 2nd degree burns to my hand?

Yes, you have a case. We have a law here in Florida called "respondeat superior" which translated into English merely says an employer is responsible for the negligence of an employee. Assuming the waitress was an employee of the restaurant (and I am not aware of any waitresses who are not employees of the restaurant) then the owner of the restaurant is responsible for any damages caused by the negligence of its employee. Just send a letter by certified mail to the manager (or whatever title that person may have) requesting the insurance adjuster for the insurance company which insures the restaurant to contact you. If after several weeks has transpired and you have not been contacted by an insurance adjuster, then go hire a good personal injury lawyer. Try to hire one who is board certified, which means they are held out to a higher standard.