Given that the domain has a real analytic boundary the construction is standard. A Schwarz reflection can be performed locally near each boundary point, which together with normal analytic continuation produces the sought extension.

(As a particular case, which however is not so relevant given the new general formulation of the question: Every biholomorphism $\phi$ of $D^2(r)^o \subset \mathbb{C}$ which has a continuous extension to $D^2(r)$ extends to a *biholomorphism* of the *Riemann sphere*. Here we use a Schwarz reflection along $\partial D^2(r)$.)