Heat the wok over medium-high heat. Add 3-4 tablespoons of coconut cream from the top of the can. Cook for 1 minute, stirring constantly. Stir in the curry paste and bring to a boil until small bubbles appear, the edges change color, and the mixture is fragrant.
Add the chicken strips and fry for 2 minutes. Add half of the remaining coconut milk, 3 tablespoons of fish sauce, and the palm sugar. Simmer over medium heat for 5 minutes.
While the curry simmers, roast the peanuts until fragrant (about 3-5 minutes), then let cool. Coarsely crush the cooled peanuts in a mortar. Stir the crushed peanuts, chilli peppers, and kaffir lime leaf strips into the curry and simmer for 1 minute. Season to taste with lime juice and fish sauce.