If the PID is designed for 110v it could hurt it to power it with 220v. The diagram that was provided above is great for what you are trying to do. It provides 220v to the heating element and 110v to the controller. If you follow the line diagram you should be ok.
There are some systems that have what's called a wild leg and it will measure 220v to ground. These set ups are rare and usually in commercial buildings. The heating element in this post is a 220v so you most likely will not hook up the neutral but to 2 phases. Both phases will be supplying 110v...
Think of it like a water shut off valve. The electrons flow thru the heating element but only when the valve is on. If the valve is off the electrons can not flow thru the heating element. If the element has two hot leads going to it I would assume it is designed for 220 or 460 volts. Turning...