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.
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.