Flush Mounted PVC Electrical Junction Box or alternative ideas to supply two sub-panels
You'll have to bore the stud anyway, so I'd just nipple between the two panels

Since you'll have to put a hole in the stud no matter what you do, I'd make that hole count as much as possible.  How? Well, we start by scrapping the junction box and using a rigid metal conduit nipple between the adjacent sides of the two panels.  This gets rid of a wire and a nailplate, as the conduit provides physical protection while serving as a grounding means provided appropriate bonding locknuts and insulating bushings are used.

It also means we can use the 60% fill rule for nipples under 24" in length, which allows us to fit 3 250kcmil XHHW-2 aluminum wires (enough for a 200A feeder!) down a 1.5" RMC nipple (which is small enough to stay within the 40% stud boring rules).

As to that feeder run to the shed...

Running the feeder to the shed in conduit is a good idea; however, you don't need the junction boxes.  I'd simply run straight out the opposite side of the panel if at all possible, then to a LB with its cover protruding from the drywall that is nippled to the LB on the outside.  If that doesn't work, you can come out the bottom then use a LR to turn the corner to the lateral run to the LB, again with the conduit body's cover protruding from the drywall.

As to the feeder wires themselves, you have two choices.  One would be to get a bunch of 1/0 Al XHHW-2 and run that for your hots and neutral with a 6AWG bare copper ground; this gives you a 125A-capable feeder, but requires a bit more labor than using a multiplexed cable in conduit.  The other option would be to use a 2-2-2-4 mobile home feeder (tri-rated USE/RHH/RHW-2) multiplex cable to get you a 90A feeder to the shed; while this compromises a bit on ampacity, it does get you the convenience of a multiplex cable, as 1/0-1/0-1/0-2 is not available in mobile home feeder and quadruplex URD can't be used for a feeder in conduit as it's made of the old, nasty AA-1350 aluminum, which isn't legal for the conductor types generally usable in conduit as per NEC 310.106(B) and 310.10.  Running SER in conduit is also a bad idea; while technically legal, it's a nightmare from a conduit fill and pulling perspective, to the point where using individual wires is a far better solution.

GO BIG OR GO HOME

The single most important thing with this installation, no matter what else you do, is to not skimp on panel sizing the way your builder did to put you in this pickle to begin with.  I would go with a 200A or 225A, 40-space or 42-space, main lug panel as a minimum for the house subpanel, using the aforementioned 250kcmil feeder wires landing on main lugs in the subpanel and a 200A breaker in the main panel to provide a shutoff for the subpanel.  In Homeline, this would require a HOM2200BB for the feeder breaker and a HOM4080L225PC with a PK27GTA field-fitted (or a HOM4080L225PGC, if you can find it for a good price) for the subpanel.

For the shed panel, you'll want a 100A or 125A, 24-space or 30-space, main breaker panel, as the main breaker serves as a local shutoff for the building.  In Homeline, this would be a HOM3060M125PC with a PK23GTA field-fitted to it, or a HOM2448M125PRB if you need an outdoor panel for the shed, again with a PK23GTA field-fitted to it. 

In both subpanels, you'll need to make sure the neutral-to-ground bonding screw is removed, by the way.  The shed will also need a grounding electrode system; two ground rods driven 8' deep and 8' apart, connected to the panel using a run of 8AWG bare copper wire, will typically suffice.

TORQUE ALL LUGS TO SPEC

One other thing you'll need to do when installing this is torque all the breaker and loadcenter lug setscrews to their manufacturer specified installation torques; this is required by 2017 NEC 110.14(D), and is a very good idea anyway lest your electrical system develop a case of the loose lugnuts...