Constuction of differential operator with specific properties

Question

Is there a way to construct a differential-type operator $\hat{D}_{\vec{x}}$ such that $$\hat{D}_{\vec{x}}\left(\frac{\vec{x}-\vec{x}^{\hspace{0.2ex}\prime}}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime}\rVert^3}\cdot\frac{\vec{x}-\vec{x}^{\hspace{0.2ex}\prime\prime}}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime\prime}\lVert^3}\right)=\delta(\vec{x}-\vec{x}^{\hspace{0.2ex}\prime})\delta(\vec{x}-\vec{x}^{\hspace{0.2ex}\prime\prime})$$ I know that the corresponding operator of the similar problem $$\hat{D}_{\vec{x}}\left(\frac{1}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime}\lVert}\right)=\delta(\vec{x}-\vec{x}^{\hspace{0.2ex}\prime})$$ is the Laplacian $\hat{D}_{\vec{x}}=\nabla^2$. Since one can write the content of the bracket as $$\frac{\vec{x}-\vec{x}^{\hspace{0.2ex}\prime}}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime}\rVert^3}\cdot\frac{\vec{x}-\vec{x}^{\hspace{0.2ex}\prime\prime}}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime\prime}\lVert^3}=\nabla_{\vec{x}}\left(\frac{1}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime}\lVert}\right)\cdot\nabla_{\vec{x}}\left(\frac{1}{\lVert\vec{x}-\vec{x}^{\hspace{0.2ex}\prime\prime}\lVert}\right)$$ I thought I could go from there, but had no luck. $\hat{D}_{\vec{x}}$ can also be a formal series of the type $f(\nabla^2)$ or similar, that's not an issue. Any help is greatly appreciated.