In general terms it is feasible.
On windows, you would need to google for “virtual audio cable”, that will lead to softwares which allow to create virtual inputs and ouputs.
Essentially, your microphone input would go to virtual input, and then that will need to split the signal to two virtual outputs.
You would set Jackaudio input to one of them, and your DAW input to the other, so whatever you play is recorded.
Same with the audio coming from Soundjack.
On Mac you probably do not need additional software, the OS is already capable of splitting the signal to virtual ouputs, google it up.