Bursting of bubbles at a liquid surface is ubiquitous in a wide range of physical, biological, and geological phenomena, as a key source of aerosol droplets for mass transport across the interface. However, how a structurally complex interface, widely present in nature, mediates the bursting process remains largely unknown. Here, we document the bubble-bursting jet dynamics at an oil-covered aqueous surface, which typifies the sea surface microlayer as well as an oil spill on the ocean. The jet tip radius and velocity are altered with even a thin oil layer, and oily aerosol droplets are produced. We provide evidence that the coupling of oil spreading and cavity collapse dynamics results in a multi-phase jet and the follow-up droplet size change. The oil spreading influences the effective viscous damping, and scaling laws are proposed to quantify the jetting dynamics. Our study not only advances the fundamental understanding of bubble bursting dynamics, but also may shed light on the airborne transmission of organic matters in nature related to aerosol production.