TEM/HRTEM can help you find the solution to your question. Before and after the desired growth reactions around the core, you can use TEM to find that.
If your material gives surface plasmon resonance (SPR) than a more simple way is to check using UV- VISIBLE spectroscopy. First check SPR for the seed nanoparticle and than check if there is any SPR shift or damping due to the shell formation on core