How can I mount a large, heavy, half-inch-thick sign on a wall WITHOUT drilling through it?
By not excluding drilling into the wall, you may have more options at hand. The first configuration that comes to mind is a modification of the "hidden shelf" design.

Drilling a hole in the studs to take bolts forming a line to support the bottom of the sign would be similar to the hidden shelf construction. A piece of trim cut with a rabbet with the lip outboard of the sign will prevent it from slipping off.

A similar, but less hefty rabbet trim on the top will secure the sign from tipping.

For the weight you've noted, it would be best to have as many supports in the wall as possible. There will be a small overlap on the sign where the rabbet prevents fall-off, but even one-eighth of an inch (3 mm) would be sufficient if minimizing intrusion is important.