Bring a large pot of water to a boil. Lightly salt and add carrots and green beans. Cover; return to a boil, then cook 4 minutes. Drain; rinse under cold water.
In a large nonstick skillet, add 1 cup of the coconut milk. Stir in the curry paste and bring to a simmer. Simmer 1 minute, until smooth. Add the remaining coconut milk, fish sauce and sugar. Stir until combined.
Bring back to a simmer, add the chicken and simmer, uncovered, for 10 minutes, turning chicken halfway through cooking. Add carrots and string beans and heat through. Turn off heat. Tear basil into pieces and stir into the curry.
While curry is cooking, soak and cook noodles following package directions.
Serve the curry and noodles, garnishing with additional basil if desired.