Set the Crockpot to low heat.
Place the boneless, skinless chicken breasts in the Crockpot.
In a bowl, mix together the coconut milk, chicken broth, diced onion, minced garlic, minced ginger, red Thai curry paste, fish sauce, brown sugar, and lime juice.
Pour this mixture over the chicken in the Crockpot.
Add the sliced carrots and chopped bell peppers into the Crockpot.
Season with salt and pepper to taste.
Cover the Crockpot and cook on low for about 6 hours, or until the chicken is tender and easily shreds with a fork.
About 30 minutes before serving, add the sugar snap peas to the soup.
Once the chicken is cooked, remove it from the Crockpot, shred it with a fork, and return it to the soup. Stir well.
Serve hot, garnished with fresh cilantro.