In this talk, I will describe our work extending combinatorial interpretations for so called toric cluster variables as was previously studied by myself and Tri Lai. In [LM 2017] and [LM 2020], most toric cluster variables were shown to have Laurent expansions agreeing with partition functions of dimers on subgraphs cut out by six-sided contours. However, the case of cluster variables parameterized by six-sided contours with a self-intersection eluded our techniques. In this talk we discuss our research rectifying this issue by using Helen Jenne’s condensation results for the double-dimer model [J 2019]. While we focus on quivers of dP3 type of Model 1 and Model 4, we anticipate our techniques will extend to certain additional cluster algebras related to brane tilings. This is joint work with Helen Jenne and Tri Lai.