We consider a particular type of $8/3$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$6$ curve $\eta$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $\eta ([0,t])$ for each time $t\ge 0$ as well as the law of the left/right $8/3$-quantum boundary length process for $\eta$.