Confusion about the EMP approach to perfect complements. Solved UMP but struggling with EMP
You already got your answer in the math forum, from the economics point of view maybe it could help that you can use the relation between the expenditure function and the indirect utility to get your solution. 
Your indirect utility of the UMP is $v(p,w)=\frac{m}{p_x+p_y}$, this provides the expenditure function for the EMP $e(p,\overline{u})=\overline{u}\ (p_x+p_y)$. Now the hicksian demand is $h(p,u)=(u,u)$. This is the case of course of the utility $u(x,y)=min(x,y)$.