This is not cut and cover. Cut and cover is when you "cut" an excavation from the ground surface down and then "cover" it back up with dirt after constructing the structure you want below ground. You have linked to an article on NATM, which is a conventional (read: non-TBM) tunneling method.
Cut and cover is what I described: excavate down, build the structure, cover it up. I am aware that there are lots of options, you can browse my comment history and see that I probably have more experience designing and constructing underground infrastructure (and tunnels in particular) than anyone else that posts on HN.
Cut and Cover is often used in transit conversations to mean any variation where you dig from the surface and not the construction method that is cut and cover. These are two different domains talking about different things that are related. Something we both need to be more aware of in conversations.
Sometimes a temporary cover is built first to minimise disruption on the surface.
Here's an old example of an umbrella bridge over Oxford Circus during the construction of the Victoria Line. There's a longer video out there of the construction of the Victoria Line that covers this in more detail.
Some of the stations on the Broadway Subway Skytrain extension in Vancouver use a similar approach, where half the road is closed and a road deck is built, then traffic is shifted over to the new road deck while excavation takes place from the side. There's some great views of this while riding the bus.