In a dry, heavy pot over low heat, cook onions alone, stirring often, 15–20 minutes until deeply softened and lightly golden. Do not add oil or butter yet.
Add niter kibbeh, garlic, and ginger. Cook 3 minutes until fragrant.
Stir in berbere, cook 2 minutes to bloom the spices.
Add tomato paste, cook 1 minute more.
Pour in chicken broth, stir to combine, and bring to a simmer.
Add chicken pieces, nestling them into the sauce. Cover and simmer 45 minutes, stirring occasionally, until the chicken is tender and the sauce has thickened.
Add hard-boiled eggs, spoon sauce over them, and simmer uncovered 10 more minutes.
Season with salt to taste and serve hot.