Connectivity question with Time Machine and two Minis

No, no. I intend to use the Time Machine back up as the device for transferring everything from one computer to the other. I backup the old computer as usual. Then I plug the external hard drive into the new computer, and I use Time Machine to restore the contents of the Time Machine backup from the external hard drive to the new computer.

I have done this before with other Macs. It has always worked. The disadvantage is the amount of time the process takes. That time is not a problem at home. With the computer at home I can let the restore function run all night.

Now that I think about it, I could probably do the same thing at the office. It would just be a simple matter of detaching the old Mac from its peripherals and then connecting them to the new Mini. PROBABLY the restore function would work overnight, too.

But the office computer is far more important than the home computer. I would like to be able to supervise the process. I would like to do so while running my business and without duplicating all the peripherals for just the one day that the process should (I hope) take.

That is where my Ethernet connection comes in. While Time Machine restores to the new Mini through USB, I would like to connect the two Minis through Ethernet. The purpose, again, is not to transfer the contents from the old Mini to the new Mini. Again, that transfer would take place through Time Machine in USB. The purpose of the Ethernet connection is solely to establish a connection between both Minis and my peripherals, like the keyboard and the monitor. I could then carry on my business in the old Mini until I have fully restored all my data into the new Mini.

Can I control two Minis linked by Ethernet in this manner?

I ended up doing a Migration into the new Mini. Fortunately the old Mini failed on a Saturday, so I could let Time Machine restore over the course of the weekend.