Melt 1/2 cup butter in a skillet over medium-low heat. Add onion; cook and stir until translucent, 5 to 6 minutes. Add 1/2 cup water and tomato paste; stir until hot, about 2 minutes. Stir in berbere, remaining 1/2 cup butter, garlic, and ginger. Reduce heat to low; cook until mixture thickens to a paste consistency, 20 to 30 minutes.

Stir remaining 2 cups water into berbere paste; add chicken. Simmer until thickened to sauce consistency, about 45 minutes.

Stir wine, cardamom, and black pepper into sauce; add hard-cooked eggs. Cook until sauce is slightly reduced, about 15 minutes more.