Place the noodles in a large bowl; cover with 1 inch of water. Let soak for 15 minutes. Drain well.
Meanwhile, in a medium pot, heat the oil over medium heat until shimmering. Add the curry paste and cook, stirring frequently, until slightly darkened, about 2 minutes.
Stir in 3 cups water and the coconut milk, brown sugar, fish sauce and crushed red pepper. Bring to a simmer over medium-high heat. Add the noodles and green beans and simmer, stirring occasionally, until the green beans are just tender, about 5 minutes.
Add the chicken and basil sprig. Simmer until the chicken is just cooked through, about 2 minutes; season with salt. Discard the basil sprig. Divide the curry among bowls and sprinkle with the sliced basil.
