Slice and prep all the aromatics. Thinly slice galangal, lemongrass, and shallots. Tear kaffir lime leaves in half. Quarter the tomatoes and slice the chicken into bite-sized pieces.
Start the broth. Bring water to a boil in a pot. Add galangal, lemongrass, and lime leaves. Simmer briefly to infuse the broth.
Add the chicken. Add sliced chicken and simmer until it's no longer pink.
Season the soup. Stir in chili paste, fish sauce, lime juice, and palm sugar. Simmer until the sugar dissolves.
Add vegetables. Add mushrooms, tomato wedges, and shallots. Simmer until just tender but still intact.
Finish with evaporated milk. Pour in evaporated milk, stir well, and turn off the heat. Serve hot!