In Thailand, tom kha gai is eaten more like a curry instead of a soup, paired with a plate of rice and spooned onto your rice before eating it. If you are looking for an authentic recipe, you are not in the right place. I can refer you over to the Kitchn for something more involved. My recipe for Thai Chicken Coconut Soup is full of shortcuts but produces a silky and aromatic soup that is a complete meal in a bowl.
I don't want the daunting task of tracking down galangal, lemongrass, and limes leaves. I personally prefer the Thai Kitchen Green Curry Paste that comes in a 4 ounce jar for just a couple of dollars. I can make this in 20 minutes flat using the basic recipe below. When I have an extra 10-15 minutes I use the "kicked up" recipe.
Basic Quick Recipe
1 tablespoon olive oil (I like California Olive Ranch evoo)