Heat the oil over medium-low heat in a large nonstick pan. Add the light scallions, garlic, peppers, and mushrooms and cook, stirring frequently, until softened, 3 to 4 minutes. Do not brown. Remove the vegetables from the pan and set aside.

Add a little more oil to the pan along with the coconut milk, red curry paste and fish sauce, and whisk together. Bring to a gentle boil, then reduce heat and simmer until thickened, a few minutes. Add the chicken and simmer, uncovered, stirring occasionally to promote even cooking, until the chicken is cooked through, 5 to 6 minutes. Return the vegetables to the pan. Stir in the lime juice, dark scallion greens, and cilantro. Taste and adjust seasoning with more lime, if necessary. Serve over rice.