At the National Sports Center for the Disabled in Winter Park, they teach never-evers to ski one-footed all the time - out of necessity. They have students who actually only have one useful foot.
The secret for those who don't already ski well is outriggers. They need to be set an appropriate length to allow the student to stand accurately, i.e., not hunched over. The 'riggers provide some of the balance platform that is otherwise missing when on a single ski. The skier can move to either edge with more confidence than they would have otherwise. In many ways, the goals are the same. Guide the left outrigger left to initiate a left turn, etc. Learn to stand accurately and fully on the foot so that, with practice, the riggers touch down only occasionally and can be shortened. Eventually, they only touch when poles would touch. These days, I don't have access to outriggers, and I do my single-footed skiing with poles.
The first few turns are the hardest, and should be done on the same terrain as that used by any other beginner. A black diamond is not the place to learn one-footed skiing.
One-footed skiing is easier with a modern shaped ski, since moving to either the BTE or LTE will cause the ski to pull into a turn, especially if the tip is loaded a little bit. I've never been able to execute a successful turn with only one ski while sitting on the tail.
As noted in other posts, balance is everything. Not only lateral, but also fore and aft.
Sideslips are a useful exercise to learn balance over the LTE. They can progress into a shallow traverse on the LTE of the uphill ski, ending in a skidded turn up the hill. This also teaches edge control and skidding on the LTE. Subtle edge control on the LTE is essential. Most of us learn to skid only on the BTE.
The traverse can progress to garlands done on the LTE. Release the edge and let the tip go downhill. Engage and let the tip pull uphill.
Then do it on the BTE. Now to release it is necessary to move downhill toward the LTE. That makes it tougher for a lot of people. This progresses from releasing and re-engaging the BTE to a full release of the BTE and engagement of the LTE to complete the turn.
When skiing on one foot, keep the other foot quiet and close to the one you're skiing on. Waving it around to keep your balance is even less effective than waving your arms around to keep your balance. If you're skiing well on one foot, it won't be obvious to any onlookers because your skiing will look very much like two-footed skiing, except that your feet might be too close together.
One of the hardest things about actually only having one ski is riding the chair lift. Do not put the foot without a ski on it down on the snow, ever! It'll pull you right off the chair. When getting off the chair, slide forward on the chair and get your shoulders, arms and hands forward to help you stand up, because you're not going to be able to use both legs. Wait to actually stand up until the ramp starts to drop so that the chair can help you stand.
For a two-footed skier, one-footed skiing can be a revelation. With two feet, they can force a turn from the back seat. With one foot, they can't. With two feet, they can recover from tipping inside too much or not enough. With one foot, recovery from such mistakes is much more difficult, especially if the skier is using poles instead of outriggers. One footed skiing demands more accurate movements.
Keep raising the bar until you can do a pure carve on either edge. Then - bumps!