Begin by rinsing the cod under cold water to remove any visible salt. Soak the cod in a bowl of cold water for an hour, refreshing the water every 15 minutes. After soaking, transfer the cod to a small saucepan, cover with fresh water and a lid, and bring to a boil. Drain and repeat the process, boiling again until the fish flakes easily. Use two forks to break the cod into small pieces and set it aside.
In a large nonstick skillet, warm oil over medium heat. Sauté onion, scallion whites and light greens, bell pepper, and Scotch bonnet pepper until they soften and become translucent, about 6 minutes. Stir in garlic, thyme, and paprika and cook for 2 more minutes.
Add the prepared cod to the skillet, cooking for about 3 minutes. Follow with tomatoes, cooking until the vegetables are tender, roughly 2 minutes.
Finally, gently fold in the ackee and heat for about 2 minutes, taking care not to overmix. Turn off the heat, mix in the dark greens of the scallions, and season with salt and pepper. Serve hot with rice and peas or plantains.