In a medium, microwave safe bowl, microwave the carrots, onion, peppers, garlic, ginger and 2 tbs of vegetable oil for 8 to 10 minutes. Stop often to stir. Pour veggies into slow cooker.
Stir the broth, lemon grass, and tapioca into the slow cooker with the veggies. Sprinkle the chicken thighs with salt and pepper and place in the slow cooker. Cover and cook on low for 4 to 6 hours.
Remove the chicken from the slow cooker and place in a large bowl. With two forks, pull into bite sized pieces.
Skim the surface of the stew for any fat with a large slotted spoon.
Heat the coconut milk in a microwave, stirring often, for 3 minutes, or until heated through. Mix in the lime juice, fish sauce, curry paste, and brown sugar. Stir into the stew.
In a bowl, steam the snow peas with the remaining 1 tbs vegetable oil in the microwave until tender, about 5 minutes. Return the shredded chicken and snow peas to the slow cooker.
Stir in basil or cilantro and serve with fresh lime wedges.