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: 
I don't think there is a contradiction here. 

In The Doctor's Wife, they *assemble* a partial TARDIS from bits and pieces of other broken ones that are lying around. They don't build anything from scratch. It's not at all inconsistent to say that the important parts have to be grown, but that you still need to fit them together.
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...).