Season chicken with salt and pepper, to taste.
Heat oil in a large stockpot or Dutch oven over medium heat.
Add chicken, shallots and garlic to the stockpot and cook until golden, about 3-5 minutes.
Stir in red curry paste and ginger until fragrant, about 1 minute.
Stir in coconut milk. Bring to a boil; reduce heat and cook, uncovered, stirring occasionally, until reduced and thickened, about 10-15 minutes.
Stir in broccolini until just tender, about 3 minutes.
Remove from heat; stir in green onions, cilantro and lime juice; season with salt and pepper, to taste.
Serve immediately with rice.
DONE!