Is it permissible for a student of knowledge to open a restaurant that serves only breakfast meals, especially if it will be opened next to another restaurant that offers the same meals, and is zakat obligatory upon it?

There is no harm in opening a restaurant next to another restaurant, and this is not considered "selling over another's sale," which is forbidden by Islamic law. The forbidden form of "selling over a sale" is when a buyer requests to annul a sale so he can sell it for a lower price, or when a seller requests to annul a sale so he can buy from him for a higher price.

A student of knowledge is not to be reproached for engaging in trade and earning a living; rather, he is to be reproached for being a burden on others. It is incumbent upon a student of knowledge to strive to accumulate wealth so that he can be independent of people, because that aids him in his religious matters.

As for Zakat, it is obligatory on the income of the restaurant if it reaches the nisab (minimum threshold) by itself, or when combined with other cash, gold, or silver, and a hawl (one lunar year) has passed on it.