I went through a similar change from PM(-T in my case) to SDE, and I can +1 the parent. The best way to make a mid career change is to find a job where the old and new role can overlap, which I how I did it as well. If you're currently working as a PM, try to get an engineering role for the same product. The engineering part will be new, but you will be able to leverage your domain expertise.
I will caution you that SDE is a job family that requires a ton of practitioner experience, especially at the higher job levels (and you likely don't want to transition to a lower level). The only way I was able to make this change was because I have a relevant college degree and because I started out my career as an SDE and always kept up to date. And even with that, it took me 2.5 years to make the job change official.
Still, don't let that caution scare you, if you really want the change and you want it for the right reasons. SDE is probably the easiest of the STEM disciplines to get into without a format education in the field.
And finally, another big +1 on the parent's comment about PM-T with technical experience being super valuable.