In a large skillet over medium-high heat, heat oil until shimmering then add chicken, onion, and garlic to skillet. Stir occasionally until chicken is cooked, about 7 minutes. Add broccoli, cauliflower, and carrots to skillet and cook 5 minutes until veggies are golden.
Stir in tomato paste, ginger, seasonings, chicken broth,and coconut milk until combined. Reduce heat to medium-low and let curry simmer until slightly thickened and veggies are tender, about 10 minutes.
Remove curry from heat and season with salt and pepper and stir in lime juice. Serve curry warm with hot rice.