Make Hamburger Patties: Whisk the Worcestershire sauce and seasonings together in a bowl. Add the ground beef, and gently combine. Form into 4 equal patties (about 1 inch thick) then make a 1/2″ indentation in the middle of each patty with your thumb or measuring spoon (this will help the burgers cook evenly). Time permitting, pop the patties the freezer or refrigerator for 15 minutes while you prep the rest of the ingredients.
Cook burgers: Heat 2 tablespoons vegetable oil in a large cast iron skillet over medium-high heat. Once hot, add the patties and cook until brown and slightly charred on one side, about 3-5 minutes, flip, reduce heat to medium and cover. Cook an additional 2-5 minutes, depending on how rare you like your burger. Remove and keep warm.
Make Gravy: To the drippings left in the skillet, melt the butter over medium-high heat. Stir the mushrooms and onions into the butter. Cook until the mushrooms are deeply golden brown, about 7-10 minutes, only stirring occasionally, pressing into an even layer in between stirring.
Reduce heat to medium; add garlic and red pepper flakes and sauté for 30 seconds. Add flour and cook for 1 minute, scraping up the browned bits on the bottom of the pan. Reduce heat to low and slowly whisk in the broth followed by the rest of the ingredients. Bring to a simmer until thickened to desired consistency; remove from heat. Season with salt and pepper to taste. If the gravy thickens too much by the time you're ready to serve, whisk in additional broth or water.
Fry Eggs: Crack the eggs in separate bowls or ramekins. Heat 1 tablespoon of oil in a nonstick sauté pan over medium heat. Working in batches as needed, add eggs to the pan on opposite sides. Cook sunny-side up, 3 to 4 minutes, until the whites are cooked and the yolks are loose. Transfer to a plate.
To serve: Divide the rice between 4 plates. Top each mound of rice with one burger patty, a generous ladle of gravy and one cooked egg. Garnish with green onions.