The outer structure is an inverted pseudo-cleft. What ... like is that is a pseudo-cleft.
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The outer structure is an inverted pseudo-cleft.
What ... like is that is a pseudo-cleft.
That is what ... like is the inverted pseudo-cleft.
The inner structure is a cleft.
Unclefted form: I like that.
Clefted form highlighting 'that': It is that that I like.
The first 'that' of the cleft form to the front, making it 'what':