  For any compact surface $\Sigma$ with smooth, non-empty boundary, we
construct a free boundary minimal immersion into a Euclidean Ball
$\mathbb{B}^N$ where $N$ is controlled in terms of the topology of $\Sigma$. We
obtain these as maximizing metrics for the isoperimetric problem for the first
non-trivial Steklov eigenvalue. Our main technical result concerns asymptotic
control on eigenvalues in a delicate glueing construction which allows us to
prove the remaining spectral gap conditions to complete the program by
Fraser--Schoen and the second named author to obtain such mazimizing metrics.
Our construction draws motivation from earlier work by the first named author
with Siffert on the corresponding problem in the closed case.
