Let $\mathbb{D}$ be the unit disk on the plane and let $U,V\subset \mathbb{D}$ be open and such that $U\cup V=\mathbb{D}$.

Is there a holomorphic map $\varphi:\mathbb{D}\times \mathbb{D}\to \mathbb{D}$ such that $\varphi(U\times V)=\mathbb{D}$?

Of course one can ask the same question for "ambient domains" other than $\mathbb{D}$.