First, find a conformal map from the half-disk to the upper half plane, making sure that the half circle is sent to the positive real half-line, and the diameter of the half-disk is sent to the negative real half-line. (Hint: look at the map $z \mapsto - \frac{1}{2}(z + z^{-1})$). Let's call this map $\Phi$.
Second, consider the Arg function (or, if you like, the imaginary part of the logarithm with a suitable branch cut making it analytic on the upper half plane). Define your Arg so as to assign the value zero to the positive half line and $\pi$ to the negative half line.
Third, having done all this, you've concocted a harmonic function $u$ on the upper half plane. Now form $u \circ \Phi$, which takes on the correct boundary values.