Step 1: Recall the momentum operator is $\hat{p}_x = -i\hbar\,\partial/\partial x$, obtained by acting on $e^{i(kx-\omega t)}$ and identifying $p = \hbar k$. Energy is the time-conjugate quantity, so by analogy we look at the time derivative instead of the space derivative.
Step 2: Write the de Broglie wave as $\psi \propto \exp\!\left[\dfrac{i}{\hbar}(px - Et)\right]$, which encodes both momentum $p$ and energy $E$.
Step 3: Take the partial time derivative:
\[\frac{\partial \psi}{\partial t} = -\frac{iE}{\hbar}\,\psi \;\Rightarrow\; i\hbar\frac{\partial \psi}{\partial t} = E\,\psi.\]
Step 4: The prefactor that extracts the eigenvalue $E$ is the energy operator $\hat{E} = i\hbar\,\partial/\partial t$. Note the sign: options with $-i\hbar$ or with $\nabla$ (spatial) are the momentum-type forms, not energy.
\[\boxed{\hat{E} = i\hbar\,\frac{\partial}{\partial t}}\]