Warm the chicken and beef broths in a saucepan until steaming; keep over low heat.
Whisk in the chicken and beef bouillon until dissolved, then keep the broth hot.
Melt butter in a medium saucepan over medium heat.
Whisk in flour to form a smooth roux. Cook, stirring, until light golden, 4–6 minutes.
Stir in onion powder, garlic powder, and black pepper; cook 30 seconds.
Slowly whisk in the hot broth, a splash at a time, until lump-free and combined.
Simmer gently, whisking often, until thick and silky, 5–8 minutes.
Season with salt. Stir in pan drippings if using. Rest 2 minutes before serving.