Heat the oil in a skillet or non-stick pan over medium-high heat, sauté the onion until it begins to soften (about 5 minutes); stirring intermittently.
Add the garlic, ginger, and cook for about 1 minute, or until fragrant; stirring frequently.
Add the chicken and brown (about 5 minutes).
Stir in the broth and cornstarch, then add the coconut milk, carrots, broccoli, red curry paste, salt, pepper, and stir to combine. Reduce the heat to medium, and allow mixture to gently simmer for about 10 minutes, or until liquid volume has reduced as much as desired and thickens slightly.
Add the spinach, lime juice, and stir to combine. Cook until spinach has wilted and is tender (about 1-2 minutes). Taste and optionally add brown sugar, additional curry paste, salt and pepper to taste. Garnish with scallions.