Add the chicken and cook for about 5 minutes, or until chicken is done; flip and stir often to ensure even cooking.
Add the garlic, ginger, coriander, and cook for about 1 minute, or until fragrant; stir frequently.
Add the coconut milk, carrots, Thai curry paste, salt, pepper, and stir to combine. Reduce the heat to medium, and allow mixture to gently boil for about 5 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 to 2 minutes. Taste and optionally add brown sugar, additional curry paste, salt, pepper, etc. to taste.