Signal events where multiple missing neutral particles are present in a final state represent challenging topologies to search for new physics at the LHC. The key to any search is the ability to separate background-like events from signal-like events. Identifying such signal-like events, and extracting their properties, is exacerbated by a lack of knowledge of the particle masses and some missing kinematic handles. We present a new method to search for open final states where the decays of the two parent particles may proceed via the same, or different, channels. For final states with weakly interacting particles, a new basis of variables can be derived using the "Recursive Jigsaw" technique. We discuss the application of this procedure by considering final states containing 2 b-jets and 2 charged leptons and missing transverse momentum. We apply the “Recursive Jigsaw” reconstruction to extract quantities sensitive to scales and angles on an event-by-event basis. The utility of this variable basis is demonstrated by studying dileptonic ttbar decays in comparison to various parameter choices for direct stop production, where each stop decays to bottom chargino, which subsequently decay to a neutralino LSP via either on- or off-shell W decay. The applicability of the variable basis is further demonstrated using resonant ttbar production through a heavy graviton.