When writing multiple papers from the same dataset, how do you handle the methods section?
There is a divide in how to approach this question based on the type of methods section, as there are two almost entirely distinct forms of methods sections in scientific publications.


"Prose" methods: This type of methods description is part of the main flow of text, and presents the methods in an explanatory fashion.  Such a methods description is expected to be either unique or else credited as being reproduced/adapted from another publication.  An example of such a paper (in particular Section VI A, B).
"Cookbook" methods: This type of methods section comes at the end of a paper, after its discussion and conclusions, and is effectively a compressed "appendix" with no narrative exposition, just an extremely terse summation of particulars.  Such a methods section is not generally subject to concerns regarding self-plagiarism (and arguable not even copyrightable!) because they are essentially just recipes: there are only so many ways to say "Cells were cultured in LB medium at 37C." An example paper with such a methods section.


For your particular case, then, if you expect both papers to be under review at the same time, then you should already be making sure that the reviewers know this is the case. The papers should likely be referencing one another as "under review", the cover letter should make the existence of the other paper clear, and you should likely submit it as review-only supplementary information.  If you are creating "cookbook" methods, that's enough to deal with self-plagiarism.  If you are creating "prose" methods, then you should explicitly note at the beginning of the section that it is shared between the two papers, thereby making an explicit declaration of shared/adapted text and defusing concerns of self-plagiarism.

Finally, if one paper ends up significantly preceding the other, such that they are not effectively co-published, then you can simply remove its references to the other publication and change the references in the second to refer to prior rather than simultaneous publication.