There does seem to be a limited choice in deep mouldings. Specially narrow and deep (deep thin gilt hockeystics that used to be available and asked for seem to have fallen out of fashion or been discontinued) so for making frame extensions ventons have one of these that I use as a general purpose job.
this one doesn't need to be hidden, although its only about 25mm deep, quite nice when painted! its also a nice 5mm front, so it sits flush to the rebate and holds the glazing in nicely.
you can see it in the pic below. I find these are very useful in general, it is actually supposed to be a moulding (rather 70s style) rather than a spacer. this is a pic from our website.

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it will fit nicely in any deep moulding....

heres that spacer, in red - does away with a mount
heres one we stained (briwax medium brown) same as the obeche frame, as you can see it is deep enough to take a plaque inside the frame.
here we used it just to space mounts off decopage, beauty is that the artpiece simply sits into the back of the wedge shaped spacer. The mount is hot-melt glued to the spacer from inside, then the lot sits into any frame you choose. Or it works without the mount. (for the cuckoo clock effect if desired

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I find we have to cut iit "upside-down" in the morso to prevent unsightly breakout at the back. Normally this would be no big deal, as the finishing and sealing can hide a multitude of sins, but its good practice to make it as neat as possible and a habit that the lads in the workshop have got into.
handy stuff!
ps. a lot of the above stuff is "self fit" as we sell them online, ( shirt frames, medal cases etc) so the plan is to make it easy and simple for the customer to fit their own items.
half a dozen stainless steel dressmakers short pins in the edge of the foamcore will hold a shirt nicely, although we obviously couldn't fit them that way in the shop. But its a different matter when the customer does it at home and pops it straight on the wall right away.
The local rugby club in particlar, have walls upon walls full of shirts in frames, and they like to have a few "fresh" ones or topical ones which they put in and take out themselves.