Heat the oil over medium-low heat in a large nonstick pan.
Add the light scallions, garlic, and jalapeño and cook, stirring frequently, until softened, 3 to 4 minutes. Do not brown.
Add the coconut milk, red curry paste, fish sauce, and brown sugar and whisk together.
Bring to a gentle boil 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. (Do not let the sauce boil, the idea is to cook the chicken gently so that it's tender.)
Stir in the lime juice, dark scallion greens, and cilantro. Taste and adjust seasoning with more lime if necessary.
Serve with Jasmine Rice.