In this talk, I will introduce the double Bott-Samelson varieties, whose cluster structures have been studied by Shen-Weng. In their work, they computed these cluster variables as principal minors of certain matrices coming from pairs of decorated flags. We will give a combinatorial approach to computing these cluster variables, via a matching construction on plabic fences which generalizes the Muller-Speyer's minimal matching construction of twisted Plücker coordinates on open positroid varieties. This is a joint work with João Pedro Carvalho.