Partial filament eruptions have often been observed; however, the physical mechanisms that lead to filament splitting are not yet fully understood. In this study, we present a unique event of a partial filament eruption that undergoes two distinct splitting processes. The first process involves vertical splitting and is accompanied by brightenings inside the filament, which may result from internal magnetic reconnection within the filament. Following the first splitting process, the filament is separated into an upper part and a lower part. Subsequently, the upper part undergoes a second splitting, which is accompanied by a coronal blowout jet. An extrapolation of the coronal magnetic field reveals a hyperbolic flux tube structure above the filament, indicating the occurrence of breakout reconnection that reduces the constraining field above. Consequently, the filament is lifted up, but at a nonuniform speed. The high-speed part reaches the breakout current sheet to generate the blowout jet, while the low-speed part falls back to the solar surface, resulting in the second splitting. In addition, continuous brightenings are observed along the flare ribbons, suggesting the occurrence of a slipping reconnection process. This study presents, for the first time, the unambiguous observation of a two-stage filament-splitting process, advancing our understanding of the complex dynamics of solar eruptions.