Place the smoked haddock fillets in a medium skillet and pour over the milk, adding a splash of water if needed to fully submerge the fish. Bring to a gentle simmer over medium heat and cook for 6 to 8 minutes until the fish flakes easily with a fork, then remove the fish to a plate and reserve the poaching liquid.

Bring a small pot of water to a boil, carefully lower in the eggs, and cook for exactly 6 and a half minutes for soft-boiled centers. Immediately transfer the eggs to a bowl of ice water to stop the cooking process, then gently peel and halve them lengthwise.

In a large deep skillet, melt the butter over medium heat. Add the chopped onion and cook for 5 minutes until soft and translucent, then stir in the curry powder and cook for 1 minute until highly fragrant.

Stir the rinsed basmati rice into the spiced onions, coating the grains in butter, and cook for 1 minute. Pour in the vegetable broth and 1/2 cup of the reserved fish poaching milk, bring to a boil, then cover and reduce heat to low to simmer for 15 minutes until the liquid is fully absorbed.

While the rice cooks, gently remove the skin from the poached haddock and flake the flesh into large, bite-sized chunks. Once the rice is done, gently fold in the flaked fish and half of the chopped cilantro, warming everything through on low heat for 2 minutes.

Divide the warm kedgeree among four shallow bowls and top each serving with two egg halves. Garnish with the remaining cilantro and serve immediately with lemon wedges on the side for squeezing.