Question: In the deleted scene where the 10th Doctor leaves Rose and the Human-Doctor on the alternate universe, the Doctor gives the meta-crisis Doctor a piece of the TARDIS to grow a new one.

BUT
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In `The Doctor's Wife`, the Doctor and Idris **build** a TARDIS console "without a proper shell"

How is this possible if as seen before, the TARDIS is grown?

Answer: 
Think Frankenstein's Monster vs. cloning. With cloning, you take a chunk of a living thing, and use it to grow a copy of that original living thing. This is similar to what the meta-crisis Doctor will do with the piece of the TARDIS.

With Frankenstein's Monster, Dr. Frankenstein put together the pieces of a human being from many separate dead people. Then rejuvenated it with lightning (in the movies, at least), it sprang to life. This is what the Doctor and Idris did. The Doctor gathered the parts, and arranged them properly, Idris provided some energy to jump-start it. That was enough to get a sort of semi-TARDIS up and running.

As TARDISes are quasi-living things, there are a variety of ways of creating them, when you're limited on resources.
The basic flight systems of the TARDIS are Dynamorphic generators (wich allow traveling trough the vortex), Dematerialization Circuit, a power source, a Time Rotor (that "conduct" the energy of the power source to the Dynamorphic Generators) and a console to control the travel. A standart TARDIS incorpore more than this, making the flight more secure and accurate. Also, the TARDIS Matrix incorporate the telephacics circuits, a Translation Matrix, and it could use the Rift Energy into the TARDIS fuel. What the Doctor does is to assemble a Console with Dynamorphic Generators, a Dematerialization Circuit and a Time Rotor; and Idris (or Sexy) use his abilities of the TARDIS Matrix to absorb the Rift Energy and power up the TARDIS.
To be said, that Junkyard Console only uses a very rudimentary adaptation Time Lord Space-Time travelling technology. It was a truly miracle that it didnt blew up in the Doctor's face....

But in one hand, you cant create a TARDIS throught a TARDIS coral, what the Doctor give them its the possiblity to CLONE the Doctor's TARDIS. Creating a TARDIS is practically imposible with eny technology that it isn't Time Lord's (it needs Block-Transfer Equations, the Prime Eye of Harmony of Gallifrey, biotechlogy, allong with tons and tons of other complicated stuff...).