Robert Bryant gives an excellent answer. Let me add some more comments.
Your question is solved by Douglas who was awarded the Fields medal due to his work on Plateau's problem. His paper, however, is a bit hard to read since it is written nearly 90 year ago. Here's a modern solution due to Jost Jost. Douglas's result on your question can be roughly described as the following:
Let $a_i=\inf\{Area(\Sigma_i)\mid \Sigma_i \text{ is a disk spanned by } \gamma_i\}$. Suppose that there's an annulus $\Sigma$ spanned by $\gamma_1 \cup \gamma_2$ such that $$Area(\Sigma) < a_1+a_2,$$ then there's a minimal annulus spanned by $\gamma_1 \cup \gamma_2$. The criterion in above is called the Douglas criterion.
A final remark. To solve this question, you cann't fix an annulus $A=\{1 < |z| < 2\}$ since conformal annuli are not conformally equivalent to each other.