Suppose $(M,J,g,\Omega)$ is a Calabi–Yau $m$-fold, $L$ a compact, immersed Lagrangian in $M$ with a transverse self-intersection point at $p\in M$ with local sheets $L_{\pm},$ and $N$ a Joyce–Lee–Tsui Lagrangian MCF expander in $T_{p}M$ asymptotic to $T_{p}L_{+}\cup T_{p}L_{-}$ and satisfying $H=F^{\perp}$. Prove that for small $\epsilon>0,$ there is a unique family $\{L^{t}:t\in(0,\epsilon)\}$ of compact, immersed Lagrangians in $M$ satisfying Lagrangian MCF, such that $\lim_{t\rightarrow 0}L^{t}=L^{0}$ in a suitable sense, and for small $t$ we have $L^{t}\approx\sqrt{2t}\cdot N$ near $p$ and $L^{t}\approx L+tH_{L}$ away from $p$.