Manipulating the left side, $2^{11} \times 6^{5}=2^{11} \times(2 \times 3)^{5}=2^{11} \times 2^{5} \times 3^{5}=2^{16} \times 3^{5}$. Since $4^{x} \times 3^{y}=2^{16} \times 3^{5}$ and $x$ and $y$ are positive integers, then $y=5$ (because $4^{x}$ has no factors of 3). This also means that $4^{x}=2^{16}$. Since $4^{x}=\left(2^{2}\right)^{x}=2^{2x}$, then $4^{x}=2^{16}$ gives $2^{2x}=2^{16}$ and so $2x=16$ or $x=8$. Therefore, $x+y=8+5=13$.