Today our lecturer said that we were gonna have "half an hour" break. Initially I thought it's a mistake and it should be "a half hour" but latter I was told that it's correct. Assuming that it's indeed correct, my question is why we put the article ("an)" after the quantifier rather than before it. (such as: "a one person" rather than "one a person")
Both half an hour and a half hour are fine. They are acting as adjectival phrases for the noun break.
Without an adjectival phrase, you would simply say:
We are going to have a break.
A break is countable (you can have one break, two breaks, or more). So, it has an article in front of it.
Depending on which version of adjectival phrase you use, it will either come before break or after it.
We are going to have a break of half an hour.
We are going to have a half-hour break.