Does such a function $f$ exist?
I think the answer is no.
My attempt: Take a harmonic conjugate of $u$, say $v'$ so that $g = u + iv'$ is holomorphic on the closed unit disk (not sure what this means). Then $v$ and $v'$ differ by a constant since they are harmonic conjugates of $u$, so $f$ and $g$ differ by a constant and thus $f$ is in fact holomorphic at 0. Therefore, no such $f$ can exist.
1) What does it mean for a function to be holomorphic on a closed set? Does it just mean it is holomorphic on an open set containing this closed set?
2) Is the last step correct where I conclude that $v$ and $v'$ differ by a constant? In particular, I don't see how this can hold at 0 since $v(0)$ is not defined (as $f(0)$ is a singularity). I have a feeling this is true, but I don't know how to justify it.