We propose a technique to generate broadband chaotic and breather signals employing an optoelectronic oscillator (OEO) comprising a phase modulator (PM) and a linearly chirped fiber Bragg grating (LCFBG). The joint operation of the PM and the LCFBG forms a broadband microwave photonic filter (MPF), which allows the OEO to generate chaotic signals and breathers taking advantage of the interplay between the broadband MPF and the time-delayed feedback loop provided by a long optical fiber delay line. The breather excitations are characterized by nanosecond chaotic oscillations breathing periodically at a significantly lower time-scale determined by the OEO large delay time. A theoretical analysis based on a modified Ikeda time-delayed model to include the effect of the broadband filtering process is provided. The analysis is verified by an experiment. The proposed LCFBG-based OEO and the possibility to control in the optical domain its broadband bandpass characteristics considering the flexibility, accuracy, and precision in FBG fabrication can find applications in chaos-based communications and in fast optical processing systems, such as random number generation, or optical processing in reservoir computing taking advantage of the intrinsic multiple time scales of the LCFBG-based OEO.