First, to answer Danielle's question and to expand on Raydot's answer. Clarified butter and Ghee can both be stored at room temperature, at which temperature they are usually liquid. But whilst ghee can be stored at room temperature for a considerable length of time, clarified butter cannot because unlike ghee it still contains water which will make it spoil at room temperature. You can extend their shelf life by refrigerating them but they will solidify as they get colder. The garlicky oil at the restaurant may well be some form of infused, clarified butter or it may be Yak butter (depending on how authentic the restaurant is). Yak butter is a traditional ingredient in Mongolian cuisine and I've heard it has quite a distinct flavour.
When the subject of infused oils come up, I always feel the need to talk about safety. So I'll add a word of caution to anyone who may stumble upon this question with the intent of making their own clarified butter or ghee, or oil, that will be infused with something like garlic.
There is a fairly widely known risk of Botulism forming in garlic infused oils but there also seams to be a perception that the risk is somehow limited to just garlic. In reality, pretty much everything is a potential vector for C. Botulinum contamination. This is because C. Bot is widely found in soil. So anything grown in, or that comes into contact with soil, has the potential to be contaminated with C. Bot. Garlic just happens to have certain properties that make it a more likely vector for Botulinum.
In order to prevent issues with C. Botulinum in infused oils, you either have to properly acidify what you're infusing the oil with, so that it's pH is low enough to prevent the C. Bot from being able to grow, or you have to heat it to a high enough temperature to kill the C. Bot. But the temperatures required to kill the C. Bot and destroy it's spores are so high, and the "hold time" so long that you'll burn garlic and possibly your oil too... or at the very least, change the flavour of the oil. You can also use dried garlic but honestly, your best bet is to make the oil, keep it in the fridge, and use it within a week.
