Amplification technique for proteins similar to PCR for DNA?
There isn't any technique with one exception. PCR generates DNA from DNA so you can establish cycles. For protein synthesis, an mRNA is a template to produce proteins. Most likely proteins you are interested in would not synthesize mRNAs, so that you can not establish cycles.

The exception I said earlier is Protein Misfolding Cyclic Amplification (PMCA). Some disease associated misfold proteins can induce misfolding of the same polypeptide which is not misfolded. So you can establish cycles of misfolding reaction. 

PS:

There are technique to synthesize proteins from DNAs or RNAs, but they are not chain reactions.
https://www.promega.com/products/protein-expression/eukaryotic-cell-free-protein-expression/
https://www.neb.com/products/e6800-purexpress-invitro-protein-synthesis-kit