The strong-field control of plasmonic nanosystems opens up new perspectives for nonlinear plasmonic spectroscopy and petahertz electronics. Questions, however, remain regarding the nature of nonlinear light-matter interactions at sub-wavelength spatial and ultrafast temporal scales. Addressing this challenge, we investigated the strong-field control of the plasmonic response of Au nanoshells with a SiO$_2$ core to an intense laser pulse. We show that the photoelectron energy spectrum from these core-shell nanoparticles displays a striking transition between the weak and strong-field regime. This observed transition agrees with the prediction of our modified Mie-theory simulation that incorporates the nonlinear dielectric nanoshell response. The demonstrated intensity-dependent optical control of the plasmonic response in prototypical core-shell nanoparticles paves the way towards ultrafast switching and opto-electronic signal modulation with more complex nanostructures.