I know PCR can be used to amplify a tiny sample of DNA in order to perform experiments. Is there a similar technique to use on a protein sample? More specifically, I'm not interested in "cutting" up the protein but simply making more of the original sample.
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.
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
It is impossible to amplify protein like PCR. DNA and RNA can undergo PCR because basepairs are complementary: A-T C-G A-U. But amino acids are not complementary; can you find an amino acid that matches glycine? Besides, proteins are not only a single polypeptide chain like DNA, proteins need to be folded after polypeptide chain are synthesized. Usually a protein molecule need more than one polypeptide chain to combine together to form a protein molecule. It can not happen that it works as PCR.
I would say impossible is a stretch. An inventive biochemist could create molecules that are "complimentary" to each amino acid similar to a Dna polymerase. Then a protein would be needed to both detect and translate this similarity into protein elongation. Unlikely, hell yes, impossible, no.