Prepare your fresh herbs by slicing the galanga into thin rounds, cutting the lemongrass into 2-3 inch lengths and bruising with a mortar and pestle or back of a chef's knife, and tearing the kaffir lime leaves into half or quarter pieces. Cut the stem off the Thai chilis and bruise with the back of a chef's knife.
Combine the chicken broth and water in a medium-sized pot and bring to a gentle boil over medium to medium high heat. Add the prepared fresh herbs and Thai chilis and allow to simmer for 10 to 20 minutes, until a the aroma of the herbs is fairly strong.
Remove the herbs with a strainer, allowing the Thai chilis to remain in the pot. (see NOTE). Add your thinly-sliced chicken, mushrooms, and coconut milk. Allow to cook until the chicken is cooked through, being careful not to overcook it.
While the chicken is cooking, season the broth with lime juice and fish sauce to taste. I generally start with 1 Tablespoon of each; however, the amounts will vary depending on the saltiness of your chicken broth and the acidity of your limes.
Once the chicken and mushrooms are cooked through and the soup is seasoned to your liking, serve with hot jasmine rice and enjoy!
Notes
You can serve Tom Kha Gai with the lemongrass, galangal, and kaffir lime leaves in the soup, as long as your guests know to eat around these herbs. I tend to take them out during the cooking process as noted in the recipe, but I save them on a clean plate and add them back to the soup if there are leftovers to save in the refrigerator. They will continue to infuse the leftover broth, making the next day's soup even more fragrant and flavorful. You can take the Thai chilis out at the same time as the other herbs, or leave them in the soup if you prefer more spice. If at the end of cooking, your soup is not hot enough, you can press the Thai chilis against the side of the pot with a spoon to release more spice.