Place the eggs in a small saucepan and cover with cold water by 1 inch (2.5 cm). Bring to a boil over high heat, then reduce to a gentle boil and cook for 9 minutes. Transfer the eggs to a bowl of ice water. Once cool, peel and set aside.
Pat the chicken dry and season with 1/2 teaspoon salt and the black pepper. Melt the ghee in a Dutch oven or deep skillet over medium heat. Add the onions and remaining 1/4 teaspoon salt. Cook, stirring often, until very soft and golden, about 12 to 15 minutes.
Stir in the garlic and ginger and cook until fragrant, about 1 minute. Add the berbere and tomato paste. Stir constantly for 1 minute so the spices toast lightly without scorching.
Pour in the chicken broth, scraping up any browned bits from the pan. Add the seasoned chicken and bring the sauce to a gentle simmer. Cover partially and simmer, stirring once or twice, until the chicken is tender and reaches 165°F (74°C) in the thickest pieces, about 20 to 25 minutes.
Stir in the lemon juice. Add the peeled eggs, turning them gently in the sauce, and simmer uncovered for 5 minutes. Taste and add salt or a little more berbere if needed. Serve hot with injera, warm flatbread, or rice.