Cook basmati rice in a large saucepan with 3 cups water according to package instructions. Set aside.

Heat canola oil in a large Dutch oven over medium heat. Season chicken with salt and pepper. Add chicken, shallots, and garlic, cooking until golden, about 3 to 5 minutes.

Stir in red curry paste and grated ginger until fragrant, about 1 minute.

Pour in coconut milk. Bring to a boil, then reduce heat and simmer uncovered, stirring occasionally, until reduced and thickened, about 10 to 15 minutes.

Stir in broccolini and cook until just tender, about 3 minutes. Remove from heat and stir in green onions, cilantro, and lime juice. Season to taste and serve over rice.