While wearing the Mitre of Holiness, you should be able to sacrifice anywhere (not just on altars, though doing it on an altar may still boost the odds of good effects), and/or when you get a reward for sacrificing or a prayer boon, you should be presented with a menu allowing you to choose from available options. (In the case of sacrificing, this may not allow you to select a specific artifact; it could be implemented as choosing between getting some gift that your god chooses, getting a point of luck, or other rewards).