Preheat the water bath to 141°F (60.6°C).
Salt and pepper the chicken then seal in a sous vide bag. Place the bag in the water bath and cook the chicken for 2 to 4 hours, until pasteurized.
For more information on the cooking times you can read my detailed article which addresses why is there a range in sous vide cooking times.
Cut off the top of lemongrass, using only the bottom 4 to 5 inches and cut into large chunks. Cut or tear the kaffir lime leaves into strips.
Place the chicken stock, lemongrass, and kaffir lime leaves in a pot and bring to a simmer. Let the flavors infuse for at least 15 minutes, and up to 30 minutes. Strain the lemongrass and kaffir lime leaves out of the soup.
Stir in the Tom Yam sauce and fish sauce and bring to a boil. Remove from the heat. Right before serving, stir in the lime and add additional lime juice, salt, or Tom Yam sauce to balance the soup.
Take the sous vide bag out of the water bath and remove the cooked chicken. Dry it off thoroughly using paper towels or a dish cloth. Quickly sear the chicken for 1 to 2 minutes per side, until just browned, then remove from the heat and cut into strips.
Place some of the sautéed vegetables in the bottom of a bowl and top with the buckwheat noodles. Add the sliced chicken and a ladle or two of the soup. Sprinkle the cilantro on top then serve.