Rub the chicken pieces all over with the lemon juice and set aside while you start the onions.
Add the onions to a large, heavy dry pot over medium heat. Cook with no fat, stirring often, for 25 to 45 minutes - they release water, wilt, and darken to a soft, collapsed mass. This is the flavor base; do not rush it.
When the onions are close to browning, add the spiced butter, garlic, and ginger. Cook 1 to 3 minutes, then stir in the tomato paste, berbere, black pepper, cardamom, and fenugreek until fully combined.
Add the chicken and turn to coat. Cook a few minutes until the surface turns opaque, then season with a little salt.
Pour in the stock and mead or wine - the meat should sit under about an inch of liquid; add water if needed. Bring to a simmer, taste for salt, cover, and cook gently until the meat is tender: roughly 20 to 40 minutes for boneless breast, up to about 2 hours for bone-in pieces. Confirm doneness with a thermometer (see below).
About 10 to 15 minutes before serving, pierce the peeled hard-boiled eggs all over with a fork, about 1/4 inch deep, and nestle them into the pot so the sauce soaks in. Adjust salt, add a splash of water if you want a thinner sauce, and serve with injera.