Rinse the rice under cold water until the water runs clear, then combine with 2 1/2 cups water in a rice cooker or lidded pot; cook according to rice-cooker instructions or simmer on low for 18 minutes, then rest 5 minutes.

In a bowl, gently mix ground beef with garlic powder, onion powder, salt, and black pepper; divide into four 3/4-inch-thick patties and press a small dimple in the center of each.

Heat vegetable oil in a large skillet over medium-high heat; sear patties 4 minutes without moving, flip, and cook 3–4 minutes more until medium doneness. Transfer to a plate and tent with foil.

Return the skillet to medium heat, melt butter, and sauté sliced mushrooms 5 minutes until golden.

Sprinkle flour over mushrooms and stir constantly 1 minute; slowly whisk in beef broth, then add soy sauce and Worcestershire. Bring to a simmer and cook 5–7 minutes until the gravy coats the back of a spoon; season with salt and pepper, then keep warm on low.

In a separate nonstick pan over medium-low heat, melt 1 tablespoon butter; crack in the eggs and cook 3–4 minutes until whites are set and yolks remain runny.

To assemble, mound hot rice onto each of four plates, top with a beef patty, spoon generous mushroom gravy over the patty and rice, and crown with a sunny-side-up egg.

Serve immediately so the runny yolk blends with the gravy for the best flavor.