Bring 6 cups water to a boil in a 6- to 8-quart pot. Have ready a metal steamer basket or bamboo steamer that fits the pot.
Meanwhile, whisk the coconut milk, peanut butter, fish sauce, curry paste, brown sugar, and lime juice in a 1-quart saucepan. Cook over medium heat, whisking frequently, until bubbly. Season to taste with lime juice and keep warm on low heat.
Cook the spinach in the water until wilted, about 1 minute. With a slotted spoon, transfer the spinach to a sieve and press on it to remove as much liquid as possible. Set aside.
Return the water to a boil. Fit the metal steamer basket or bamboo steamer into or over the pot. Arrange the broccoli and carrots in the steamer, cover, and steam until just tender, about 6 minutes. Remove the steamer from the pot.
Add the chicken and 1 tsp. salt to the water. Turn off the heat, cover, and let the chicken poach until cooked through, about 5 minutes.
With a slotted spoon, transfer the chicken and vegetables to a platter, season to taste with salt and pepper, and serve with the sauce.