Heat 2 to 3 tablespoons coconut oil in a large skillet over medium-high heat.
Add the diced onion and sauté, stirring occasionally, until it begins to soften, about 5 minutes.
Add the diced chicken to the skillet and cook, stirring and flipping often, until cooked through, about 5 minutes.
Stir in the minced garlic, ground or fresh ginger, and ground coriander; cook until fragrant, about 1 minute.
Pour in the coconut milk, then add the shredded carrots, diced zucchini, Thai green curry paste, salt, and pepper; stir to combine.
Reduce heat to medium and simmer gently, stirring occasionally, until the sauce has reduced slightly and thickened to your liking, about 5 minutes.
Stir in lime juice and chopped cilantro; taste and adjust with additional curry paste, salt, pepper, or sugar if desired.
Serve warm over jasmine or basmati rice or with naan, if desired.