What are my options for framing a thicker than average piece of artwork?
1cm isn't really "deep" for artwork. You won't find a photograph frame to fit it, but pretty much any frame for a canvas will be deep enough. You should be able to find these in arts / crafts stores that have a lot of frames. Often department stores will have only photo frames, so you'll probably have to look in more specialized shops.

Otherwise, any frame shop will have a ton of frame styles they can custom size for you. This will be a more expensive option, but you can get the artwork installed in the frame you select with very little work on your part.