Hopefully this will explain what the problems are in doing this with a French cleat (an American term, or in English terms a "split batten"). A split batten is made by bevel rip sawing a wider piece of timber, sau a 3 x 1in planed piece to produce two pieces with an angled meeting face:
You can purchase ready-made split battens from some trade timber merchants, but by no means all. I generally prpare my own on site with a portable table saw - this is not really a job for a hand held saw (not accurate enough). Obviously this needs a certain depth to accommodate the split batten, so if the item is less than about 70mm deep (110mm with a recessed split batten - see below) it just won't work.
The piece in red is screwed to the rear of the panel or item you want to hand on the wall, the other piece (coloured yellow) is fixed to the wall. The problem is that these will jack the item off the wall by the thickness of the cleat (generally 20 to 22mm as they finish at that size when the timber is planed). Normally when we are using these it is to hang flat panels, such as removeable fascia panels, wall cabinets or T-slot boards (think about the sort of slot walling you see in shops) so the top and bottom edges won't be visible, meaning that this offset isn't an issue. You generally suspend panels on two or more parallel split battens to allow you to plumb up the pnels.
The other issue with split battens is the the wall they are going onto has to be dead flat side to side, which of course walls never are. If the walls aren't dead flat the batten on the wall will conform to the wall and you'll struggle to get the piece to hand on the wall - so you end up using a long straight edge (such as a long spirit level) to allow you to check whilst you pack the batten off the wall to make it flat.
The problem, as I said is that split battens jack the item off the wall, so if you want the item to site flush against the wall you need to recess the the battens into the back of the mantle piece with a router - a lot of work. The same goes for the slimline metal split batten used in shop fitting - these are shallower (about 4mm), but still require recessing (and they are very sensitive to the flatness of the wall they are going onto) - so unless you recess them (as the OPs photo clearly shows) the mantle you are hanging will not pull flush into the wall
Above: This cleat is clearly recessed in a rebate to make the connection to the wall flat
There are other issues with using cleats in this case, but hopefiully this explains why they should be discounted. Split battens have their uses - just not here. Resin anchors are actually way easier to get right IMHO
