In short, we could, but the cost would be incredible.
All vertebrates are animals that develop from a series of segments, with a vertebra at the core. In our time from eel-like fish, we've specialised these segments so, for example, we have ribs on the vertebra corresponding to the rib cage.
To support arms and legs, specific vertebra have become highly specialised in the form of hips and shoulders.
Gills are composed of a series of gill arches, one on each vertebra in the neck area. These structures have (in eels) a lot of blood vessels to carry the blood that needs reoxygenation.
An interesting thing happened as the eel-like creatures differentiated, evolved jaws and ultimately ended up as mammals and humans: nature co-opted the specific vertebra that had these gill features and turned them into jaws and ears and a variety of other features in the head and neck. For example the tiny bones in your ear were once fish jawbones which were previously one (or more) gill arches.
The stupendously complex anatomy in this area comes from all the short-term 'decisions' evolution took to make all the magnificent creatures that inhabit the earth.
For example the nerve that connects the brain to the larynx (the recurrent laryngeal nerve) emerges from a vertebra high up in the neck, decends down under the aorta in the chest and then back up into the neck to the larynx. In the giraffe, the nerve is many meters long, even as it's direct path could be a few centimeters. The reason is that the heart used to be close to the gills in fish and sharks. As the heart moved in land animals, the nerve was caught in a loop around the critical aorta and it was 'pulled' along for the evolutionary ride.
So, in order to turn your gills back on, you need to unprogram 450m years of evolution of the structures you call your head, face and neck.
I'd recommend 'Your inner fish' by Shubin - it's a wonderful read and explains this in far more detail that I can manage.