Why is $H^1(X, \mathcal{O}) \neq 0$ for $X = (B(0, 1)\times B(0, 2)) \cup (B(0, 2)\times B(0, 1))$?

Question

In this MathOverflow answer, David Speyer says that

\begin{align*} X &= \{ (z,w) \in \mathbb{C}^2 : (|z|, |w|) \in [0,1) \times [0,2) \cup [0,2) \times [0,1) \}\\ &= (B(0, 1)\times B(0, 2)) \cup (B(0, 2)\times B(0, 1)) \end{align*}

is an example of a contractible manifold such that $H^1(X, \mathcal{O}) \neq 0$. I can see that $X$ is contractible, but I don't know how to show that $H^1(X, \mathcal{O}) \neq 0$. Can we explicitly write down a $\bar{\partial}$-closed $(0, 1)$-form on $X$ which is not exact or do we need to use some sheaf cohomology?