Place the oil in a large 6-8 quart sauce pot over medium heat. Once the oil is hot, sauté the onions, minced garlic, and ginger for 3 minutes to soften. Stir in the chopped bell pepper and carrots.
Add the chicken breasts, coconut milk, 3 tablespoons green curry paste, chicken broth and fish sauce to the pot. Raise the heat to high and bring to a boil. Once boiling, reduce the heat back to medium, then simmer for 15 minutes.
Meanwhile, drain and rinse the konnyaku noodles.
Using tongs, remove the chicken breasts from the pot. Use a fork and the tongs to shred the chicken. Then place it back in the pot. Stir in the konnyaku noodles and the broccoli florets. Taste, then add 1-3 more tablespoons green curry paste if desired. Taste again, then salt and pepper as needed.
Garnish with fresh basil leaves and serve warm.
NOTES: Green curry paste varies greatly in spice level and sodium, based on the brand you buy. Start with a small amount of curry paste and add more at the end. You can find green curry paste and fish sauce in the "international foods" section of most grocery stores.