You try to do thing X. You sit down and start doing it. You discover there's a dependency tree, where you need to do A, B, C, and D. The idea is, instead of doing X, now you throw your efforts to do X away and do one of A, B, C, or D instead. Possibly applying this recursively when you find C requires G, H, and I. You throw X away but keep the discovered dependency tree.

As you do this more and more often the dependency trees will become more clear until eventually you can generally skip the "sit down and try to do X" step, or at least come close.

An example is, you had code that worked on a local server but now you want to run it in the cloud, "natively". You sit down to start doing the conversion, but discover that you had some files you wrote to disk, and to be "cloud native" they ought to live in S3. You stop the big-bang "native cloud" conversion. You work on abstracting away what you do with the files into some sort of interface (whatever your local language calls it) which backs to something that implements that interface on a local file system. (In this particular case, there's a good chance someone has already written this abstraction layer and you just need to go pick it up, but it's not out of the question to write it yourself either; often the price of the two options are comparable.) At all points in this process you can ship the results, because even if it's ugly behind the scenes due to a half-done conversion, and you can, say, read files but not write them through the new interface, it's still shippable. And you can ship it when the only implementation of the interface is to the file system. But later on, you can implement something that backs to S3 and slip it in to the interface, and even though it's likely you'll still find a thing here and a thing there that need to be tweaked, it's far less disruptive than if you had stuck with the initial "make a big PR to convert to cloud native all in one go".

It is also more work. Although perhaps less "more work" than you might anticipate. A lot of the process of discovering what surprisingly deep dependencies on file system behaviors you had and untangling all of them is really the same in both cases. The saved effort of writing an explicit interface and jumping straight to the new thing can be easily lost in the additional effort of trying to juggle the intermediate phases (which can get quite complex) and feature flags and the bugs you introduce in the process. Even if you don't have a "Mikado" philosophy, you may well still find the process I outline in the previous paragraph is the one you adopt anyhow! In which case, why not do it independently and isolated, without a big bang?

I haven't thought "Mikado" explicitly in my refactoring plans, but really, the only "big bang" rewrite I'm willing to undertake is when my back is forced against the wall and I simply have to switch languages. Though there can still be some value in trying to straddle some services across in the meantime, even that is itself often a rather large rewrite on both sides. Otherwise, there is almost always some sort of path to doing your rewrite incrementally, one step at a time, without ever stopping main development.

If you look at the graph of "value obtained over time", it's really a no brainer to adopt this approach, too. Generally these incremental refactorings also have intermediate value they deliver. Shipping that value now means you start benefiting from it before the big bang cutover. It also means you can stop at any time and just pocket the gains you've made, with no risk of having all this work go to nought. The big bang rewrite needs to be a desperate last-ditch effort, not the first thing you reach for.

Honestly, they generally aren't even as fun as developers anticipate; they may be more fun than an incremental refactor in the first month, but the eternal grind of discovering some corner case that the old code base handled and you completely whiffed on and now you need a major rearchitecting gets to be not fun in the third month and every month subsequently. Over the course of a full project I think the incremental refactors are actually more fun.