Given the energy functional $$E[\Psi] = \frac{\langle \Psi \vert H \vert \Psi \rangle}{\langle \Psi \vert \Psi \rangle},$$ its functional gradient is $$\frac{\delta E[\Psi]}{\delta \langle \Psi \vert}=\frac{H\vert \Psi \rangle -E[\Psi]\vert \Psi \rangle}{\langle \Psi \vert \Psi \rangle}.$$
I do not understand how to obtain this expression. What it the rule to evaluate a functional gradient of the function $E[\Psi]$?
OP's formula seems to be the natural functional generalization of the partial derivative $$\frac{\partial E(z,z^{\dagger})}{\partial z^{\dagger}}~\stackrel{(2)}{=}~\frac{(H-E(z,z^{\dagger}))z}{z^{\dagger}z},\tag{1}$$ where $$E(z,z^{\dagger})~=~\frac{z^{\dagger}Hz}{z^{\dagger}z},\tag{2}$$ and where the variables $z$ and $z^{\dagger}$ are treated as independent.