How to line a tart ring for a perfect tart crust
TL:DR; don't chill the pieces after you cut them. Assemble it straight away.

Long version:

I think what's happening is that chilling after cutting the pieces will dry the surface, making it harder for them to stick together when assembling.

The point of putting the dough in the fridge is to help stabilise temperature, allow sugar and other solid stuff (salt, leavening if using) to dissolve, and most importantly for gluten to relax. But the fridge is also quite good at removing humidity from your products, that's why you cover it in saran wrap.

If you put the cut pieces back in the fridge, the very high surface to volume ratio will inevitably cause an undesirable moisture loss. Dry edges will not join - I've seen recipes suggest using wetting the edges when assembling similar things (with different doughs), but it should not be necessary.

The recipe itself looks fine (I would add a pinch of salt, but it does not make a difference for your purpose). Pastry flour could help, but since you're already cutting it with almonds it should not be a big issue.

Temperature is also a factor. At my latitudes, temperatures in this season (45° N, august, ~30 °C during the day) make it very difficult to make a good pate sucree in a kitchen unless there's air conditioning going on at full blast. Butter will begin melting much earlier, making it much more difficult to avoid overworking.

Here is how I would change the procedure:


Assemble the dough in the processor as per recipe
Chill it ideally 4 to 24 hours. 30 min is too little. 3 days is definitely too much.
Take it out of the fridge, knead it a bit to make it workable (you can also beat the dough packet with a rolling pin. Really!)
Roll it, cut the pieces, assemble it straight away
The side on top of the base version should be easier to work with than the base inside the sides, since the dough will contract during baking. The other version could have a slightly better look when finished though.
Use your fingers to make sure the join is... joined. The warmth of your hands should convince the butter to yield very quickly. You should not be able to see any seam.
Alternatively, I've seen pastry chefs completely avoid working the joint when using micro dotted rings as you're doing, instead applying softened butter with a brush like glue to avoid the kind of separation you're experiencing. 
Back in the fridge, turn on the oven
Bake it as per recipe.