Weyl transformation of Dirac equation

The Dirac Equation is given by $$\left(i\gamma^\mu\partial_\mu- \frac{mc}{\hbar}\right)\Psi_D = 0,$$

where $\gamma^\mu$ are the Dirac $\gamma$-matrices and $\Psi_D$ is a Dirac spinor. I would like to find the transformation $U$ such that the two-component Weyl spinors $\Psi, \hat{\Psi}$ solve the equation $$i\left( \begin{array}{cc}0 & \partial_0+\vec\sigma\cdot\vec\nabla \\ \partial_0 - \vec\sigma\cdot\vec\nabla & 0\end{array}\right) \left(\begin{array}{c}\Psi\\\hat{\Psi}\end{array} \right) - \frac{mc}{\hbar}\left(\begin{array}{c}\Psi\\\hat{\Psi}\end{array} \right) = 0$$

if $\Psi_D = U \left(\begin{array}{c}\Psi\\\hat{\Psi}\end{array} \right)$ solves the Dirac equation. Could anybody show me how to derive the tranformation matrix $U$? I read everywhere that

$$U = \frac{1}{\sqrt{2}}(1-\gamma^5\gamma^0),$$

but obviously, I don't know how to arrive at this.

Your second equation still is the Dirac equation in the chiral (Weyl) basis of the gamma matrices, which uniformizes $\gamma^0$ with $\gamma^k$, and is diagonal in chirality, $\gamma^5$,
$$\gamma^0 = \begin{pmatrix} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} -\mathbb{1} & 0 \\ 0 & \mathbb{1} \end{pmatrix},$$ and all you have to do is plug in the gamma matrices.
I assume you then want to go to this Weyl basis from the conventional Dirac basis, $$\gamma^0 = \begin{pmatrix} \mathbb{1} & 0 \\ 0 & -\mathbb{1} \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix}.$$
The unitary similarity transform of the Dirac basis $$\gamma_W^\mu=U^\dagger \gamma_D^\mu U, \qquad U=(1-\gamma_D^5 \gamma_D^0)/\sqrt{2}= \frac{1}{ \sqrt{2}}\begin{pmatrix}\mathbb{1} &\mathbb{1}\\ -\mathbb{1} & \mathbb{1}\end{pmatrix}$$
does the trick, as designed, $$U^\dagger\left(i\gamma_D^\mu\partial_\mu- \frac{mc}{\hbar}\right)U ~ U^\dagger\Psi_D = 0,$$ providing the chirally decoupled equation in the Weyl basis (your 2-2 vector) you wrote down explicitly.