Place onions in a large saucepan over medium-high heat. Cook, covered, until starting to brown, about 5 minutes. Add vegetable oil and garlic paste; cook until fragrant, about 5 minutes. Stir in tomatoes; cook until softened, 5 to 7 minutes.
Stir 1/2 cup water, dried beef, and berbere seasoning into the saucepan. Reduce heat to medium and cook until flavors combine, about 10 minutes. Continue cooking, adding 1/3 cup water every few minutes, until beef is softened and sauce is thick, about 20 minutes. Stir in kibbeh and salt.
Tear 1 injera into palm-sized pieces; fold gently into the sauce. Serve beef and sauce over whole injera.