Bring a large saucepan of water to a boil over medium-high heat. Using a slotted spoon, carefully lower eggs into water one at a time. Immediately set a timer for 7 minutes and cook eggs, maintaining a low boil. Carefully transfer eggs to a large bowl of ice water and let cool until just slightly warm, about 2 minutes. Gently crack eggs all over and peel, starting from the fat end containing the air packet. Gently dry eggs with a clean dish towel.

While eggs cook, purée ginger, garlic, yogurt, cardamom, cinnamon, pepper, and half of onion in a blender until smooth; set aside.

Heat ghee in a large nonstick skillet over medium-high until it sizzles as soon as an egg hits the skillet, about 1 minute. Fry whole eggs, rotating twice to get golden brown and crispy on 3 sides, about 5 minutes total. Using a slotted spoon, transfer eggs to a plate, leaving ghee behind; generously season eggs with salt.

Add apricots to skillet with remaining fried onions and cook over medium-high heat, stirring frequently, until just starting to blister, about 1 minute.

Add reserved yogurt purée and cook, stirring frequently and smashing against sides and bottom of pan (the yogurt paste will begin to stick to pan and darken—this is what you want; just keep stirring and smashing until you get there), until yogurt cooks down and purée is the colour of gravy, 5–7 minutes. Add bay leaf and ½ cup water to pan and scrape up any browned bits. Reduce heat to medium, add coconut milk, and season with salt.

Bring to a simmer and cook, stirring occasionally, until flavors have melded and sauce is the consistency of gravy, about 1 minute. Squeeze in juice from one-quarter of lemon; pluck out bay leaf. Return eggs (but not the frizzled onion) to sauce and cook until heated through, about 1 minute.