My favorite surfaces to paint on are cradled wood panels and flat hardboard panels so I thought I would share some ideas on how to frame this type of artwork.
Many framing companies make what are called "floater frames" (or "tray frames" in the UK). Here's a video by Ampersand that demonstrates how these frames can be used with cradled and flat panels.
This video demonstrates how to mount flat panels in a floater frame. I would recommend using this method not just for flat panels but also for cradled panels that have a thin profile (such as the ones I use that are 3/8 inch deep). Using screws to mount thin panels may cause the wood to split or worse, the screws could come through the surface of the artwork. This video also shows how the panel can later be removed from the frame if necessary.
A Few Options
Below is a list of brands and retailers that offer floater frames: