Best Panel Surface for Small to Medium sized Egg Tempera

As you note, Egg Tempera is a somewhat brittle paint, due to its high PVC (solid pigment particles don’t bend) and non-polymerizing, egg oil (which stays fluid to some degree and moves around; egg oil is what imparts plasticity to ET paint, so as it moves around/leaves the paint, ET becomes more brittle). 
For these reasons (and more – but I’ll try to keep this short) egg tempera does best on a rigid support.    Wood-based panels are most often used.  They can be either solid or engineered (plywood, hardboard, fiberboard).  They are considerations, pros and cons to each choice – see the “Rigid Support” page on the resource section of MITRA.   
Thin panels are more prone to flex than thick ones (i.e. 1/8″ hardboard is much less rigid than 3/4″ plywood).  Large panels more apt to flex than small ones (1/8″ hardboard at 5 x 7″ is more rigid than one at 14 x 18″).  There are other factors – the more layers of paint, the less flexible the paint film is; large sizes pigments (historical, native earths) make less flexible paint films than modern, tiny micron size pigments.  I don’t mean to make this complicated but, in fact, there are a lot of factors to consider.
Anyhow, depending on the above, at a certain point a panel may need to be braced.  There are different ways to do this, and different thoughts on what works best.  Glue strips of wood to the back of a panel (as the panels you recommend) is one option & can work well….then again, two different wood-based materials (thin, plywood panel and thick, solid wood strips on back) will absorb moisture at different rates, expand/contract at different rates, and that stress can travel into the gesso and paint layers and potentially cause them to crack.  For this reason, there are some people who believe bracing introduces, long term, more problems than it solves.  The same is true of aluminum bracing – nonabsorbent aluminum won’t move at all in response to RH, the wood elements will – that conflict can, over time, cause problems.  So bracing is a short term solution that can, under good conditions, work in the long term….but then again, may not.  
My own preference, instead of bracing, would be to use a very high grade plywood (the best you can find – i.e. a good cabinet-grade birch) in the thickest dimension you can afford to work with, 1/2 to (even better) 3/4″.  But I appreciate that, at large sizes, this can make for a heavy painting.  And it’s getting increasingly difficult to find really good quality plywood, without voids or lower grade filler woods – but they are out there.   
Other factors that increase stability in a panel: Coat back and front with equal number of gesso layers (this is important to minimize warping).   Paint the back with house paint to prevent RH from entering in.  Frame with a sturdy frame.  All these things help a panel to stay planar. 
Whatever type of wood-based panel you use, if it’s got a wood grain pattern, you should first coat it with cloth (linen, cotton) before gessoing – without cloth, the grain pattern can telegraph through the gesso and cause cracks in the paint.   Fiberboard panels, which don’t have a grain patter, don’t need a cloth layer.
Koo Schadler​