Let $G, H$ be finite graphs without loops or multiple edges and $K_n$ denote the complete graph on $n$ vertices. If for every red/blue colouring of edges of the complete graph $K_n$, there exists a red copy of $G$, or a blue copy of $H$, we will say that $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the smallest positive integer $n$ such that $K_{n} \rightarrow (G, H)$. Star-critical Ramsey number $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)$. In this paper, we will find $r_*(K_{1,n}, K_{1,m}+e)$ for all $n,m \geq 3$.