Can't government experiment on price in the case of a monopoly? In the lecture given by Jon Gruber at MIT, he talks about the governmental regulations on monopolies, and he states that

(in short)
for government to regulate a monopoly, it needs to know the price of
that good in the case of a perfectly competitive market.However, in
reality, the government does not know the demand and supply curve.
Nevertheless, to get the demand curve the government can do
experiments on the consumer and get the demand curve.

If so, can't the government do experiment on the monopolist ? I mean, it can set a price first, and then change the price. Doing so, observing how much the firm produce, we can learn the supply curve of the monopolist. Of course, this would in some sense be not logical, but in the long run, you the government could set all the monopolies in the optimal price.
Moreover, if we look from a broader picture, this can be accomplished more easily by cooperation of different governments on collecting the data.
If I understand your question correctly, you are proposing that the government sets prices and observes the behavior of market actors to elicit the demand and supply functions.

I see two possible concerns: 
Unless the monopoly is obligated to serve all customers, the market may not be in equilibrium, full demand may not be observed.
Similarly the government would have to buy any unsold goods, otherwise
a too large $p$ could result supply being larger then demand, resulting in a loss for the monopoly.

The other concern is that the monopoly's behavior significantly affects the government's observations. It could potentially benefit by behaving suboptimally now and earning more in the regulated future. One way it could do that is to make the government believe it has higher costs by supplying lower quantities given any price $p$.
-While I was writing this answer, @Dan has posted the same idea in a comment.