Heat 1/2 cup butter in a skillet over medium-low heat. Cook and stir onion until translucent, 5 to 6 minutes. Add 1/2 cup water and tomato paste; stir until hot, about 2 minutes. Stir in remaining butter, berbere, garlic, and ginger. Adjust heat to low and cook until mixture thickens to paste consistency, 20 to 30 minutes.
Stir remaining water into berbere paste; add chicken. Simmer until thickened to sauce consistency, about 45 minutes.
Stir white wine, cardamom, and black pepper into the sauce; add hard-boiled eggs. Cook until sauce is reduced slightly, about 15 minutes more.