We show that in integro-differential delayed dynamical systems, a hybrid state of simultaneous fastscale chaos and slow-scale periodicity can emerge subsequently to a sequence of Hopf bifurcations. The resulting time trace thereby consists in chaotic oscillations breathing periodically at a significantly lower frequency. Experimental evidence of this type of dynamics in delayed dynamical systems is achieved with a Mach-Zehnder modulator optically fed by a semiconductor laser and is subjected to a delayed nonlinear electro-optical feedback. We also propose a theoretical understanding of the phenomenon.