It seems Hamantaschen cookies unfolding and leaking is not an uncommon problem. Of course the flax egg substitution can very likely also contribute to the unfolding but here are some tips from someone who has been baking Hamantaschen cookies for a while.
Tips for Creating Perfect Hamantaschen
1) Find a great dough recipe. Often, the problems people have with
baking hamantaschen can be traced to an inferior dough. Here are two
foolproof, tasty dough recipes that I highly recommend. They are easy
to handle and shape, and they provide great results when baked:
Dairy Free Hamantaschen Dough
Buttery Hamantaschen Dough
Of course you can use your flax egg substituted in for the eggs that the recipes ask for.
2) Roll your dough out to 1/8 inch thick (or less). You want your
dough to be as thin as possible, while still being thick enough to
maintain the cookie’s structure. 1/8 inch seems to be the magic
number; sometimes I roll mine out even thinner than that. For a more
doughy texture you can roll it thicker, but remember– the thicker the
dough is, the harder it will be to handle and shape. Thick dough is
also more prone to opening/spreading in the oven.
3) Use a thick filling that won’t run/weep from the cookies while
baking. Knowing the proper consistency of a hamantaschen filling takes
experience, because each type of filling is slightly different.
Poppyseed filling has a very different texture than fruit filling, for
example. A good filling should be somewhat thick so that it doesn’t
run.
4) Cut your hamantaschen dough in 3-inch circles (or larger) before
filling and folding into triangles. Anything smaller than 3 inches
will be difficult to fold around your chosen filling.
5) Most fillings can be chilled before using to fill hamantaschen.
I’ve found that fruit, poppy seed, and cream cheese-based fillings
tend to be easier to work with when they’re chilled in the
refrigerator. The chilling process thickens the fillings and makes
them less sticky, which makes them easier to handle with when you’re
assembling your hamantaschen. Not all fillings are helped by
refrigeration, however– particularly chocolate-based fillings like
Nutella, which will harden with prolonged refrigeration. Check your
filling recipe to see if refrigeration is recommended.
6) Do not overfill your hamantaschen. Use 1 teaspoon of filling per
hamantaschen cookie. Do not use more than 1 teaspoon. However tempting
it might be to put lots of delicious filling in the middle of your
cookie, using more than 1 teaspoon can cause your hamantaschen to
spread open and leak in the oven. 1 teaspoon is plenty, especially
when you cut your dough circles to 3 inches… it’s the perfect amount
of filling.
7) Fold your triangles the right way! Using the proper folding method
will help your hamantschen hold together and create a beautiful shape.
Please that the sides are folded down so that each "side" has an "end" that is above and an "end" that is below in the fashion of folding the top of a box. This will help in preventing the cookies from unfolding.
Also note in Step 5, to pinch each of the 3 corners gently but firmly.
Source: http://theshiksa.com/2012/03/01/how-to-make-perfect-hamantaschen/
Welcome to Seasoned Advice first I will point you to an article in our blog Silpat, Parchment Paper or Plain Baking Sheet. @KatieK contrasts these cooking surfaces.Look at the instructions for time and temp. Are you thoroughly Preheating? You pictures indicate that you are using parchment, but you don't mention time/temp.
My first inclination based on what I see is that you are baking them too long at too low a temp, and not fully preheating. If you raise the temp and shorten the time the cookie will not spread as far. With 'scoop' cookies you expect the dough to spread some, but you are getting too much spread. The higher temperature will cause the cookie to form a solid (ok, solidish, it should be soft but 'strong enough' to hold form) This will also give you better "bottoms" as they will bake faster as well.
Another thing you might be doing is opening the oven door too often, this releases a lot more heat than you think, extending the correct baking time and slowing the process. I know it is sometimes hard to resist peeking, try to not open the door till you are at least 75% through the projected time.
Best Answer
The rest period hydrates the starches in flour, giving the dough a firmer and more workable texture (there is some very minor gluten development, but its mostly the expansion of the starch bundles with water). In many cookies, the flavors will also mature and improve, especially with cocoa in the recipe.
In many recipes, the cooling from refrigeration is itself part of the point--doughs may be easier to roll and cut when they are cool due to chilling of the fat. Most dietary fats are waxy, which means they don't have a single set freezing/melting point, but rather get more viscous or eventually firmer then hard as temperature drops.
For example, linzer cookie dough is very frangible, and the chilling and hydration make rolling and cutting much easier.
Generally, most of this affect is going to happen in the first 8 hours or so. The overnight thing is simply for convenience in most cases. In fact, 4 hours is often enough, although obviously not in your case.
If the major effect is chilling in your recipe, by flattening the dough to a disk, or lengthening it to a log (thus increasing surface area) compared to a ball, it will chill more rapidly. You mentioned this is a high fat recipe--if that fat is butter (as it so often is with cookies), that may help.