Cold smoke the chicken first. Set up a BBQ with a small amount of wood chips. Once the smoke is going, place the chicken breast on the cool side of the grill away from direct heat and cover. Leave for a minimum of 2 hours, or overnight in the fridge for a deeper smokiness. The chicken stays raw, which is fine. It will cook in the soup. Alternatively, grill or BBQ the chicken breast over direct heat until charred and cooked through, then slice into rough 2cm pieces. This gives a different result but still works well.
Blanch the squash. Bring a small pan of water to the boil, add the squash pieces and simmer for 3 to 4 minutes until just beginning to soften but not cooked through. Drain and set aside.
Build the soup base. Combine the chicken stock, 100ml of the coconut cream, fish sauce, salt, galangal, coriander roots, bruised chillies, lemongrass, kaffir lime leaves and Thai shallots in a medium saucepan. Bring to a simmer and let it bubble gently for 5 minutes. The broth should smell deeply aromatic at this point: floral from the kaffir lime, sharp from the galangal. If it still smells raw and faintly sweet, give it another couple of minutes before moving on.
Add the par-cooked squash, mushrooms and cold-smoked chicken pieces to the simmering broth. Cook gently for 4 to 5 minutes until the chicken is just cooked through and the mushrooms have softened. Don't let it boil hard or the chicken will tighten.
Stir in the remaining 200ml coconut cream and the sweet basil. Add the lime juice. Taste now: the soup should be properly salty from the fish sauce, with the lime sharp at the back and the chilli building slowly underneath. If the coconut sweetness is dominating everything else, the lime is what brings it back. If it tastes flat, it needs fish sauce before anything else. Don't adjust casually at this stage. This is where the bowl is made.
Divide between two deep bowls. Spoon nahm prik pao over the top at the table. Serve with steamed jasmine rice alongside.