Add the finely diced onions to a large Dutch oven over medium heat with no oil and cook, stirring frequently, for 20–25 minutes until they release their moisture and begin to break down into a thick, paste-like consistency. Once reduced, add the olive oil and 2 tbsp of butter, then continue cooking for another 10–15 minutes until the onions are deeply softened and lightly caramelized.
Stir in the East African spice blend, grated garlic, and ginger, and cook for 2–3 minutes until fragrant. Add the tomato paste and cook for another 2–3 minutes to deepen the flavor and remove any raw taste. Season the chicken with 2 tbsp lemon juice and salt, then add it directly into the pot, turning to coat it completely in the onion and spice mixture. Pour in the chicken broth and bring to a gentle simmer, then cover and cook on low for 35–45 minutes, stirring occasionally, until the chicken is tender and the sauce is thick and rich.
Stir in the remaining 2 tbsp butter for added richness, then add the hard-boiled eggs, nestling them into the sauce. Simmer uncovered for another 10–15 minutes so the eggs absorb the flavor and the sauce thickens further.
Finish with an additional squeeze of lemon juice if desired, then serve warm.
