Preheat the oven to 350°F (180°C). Grease the inside of the ramekins with the butter. Place the ramekins on a rimmed baking sheet pan; set aside.
In a medium bowl, gently mix together the ground beef, 1½ teaspoons of the Worcestershire sauce, 1 teaspoon of the onion powder, 1 teaspoon salt, and ½ teaspoon pepper. Divide into 4 equal portions, then form each into a ⅓-inch-thick patty, about 3 inches in diameter.
Break 1 egg into each of the prepared ramekins. Pierce each yolk with a knife to allow it to spread. Season with salt and pepper. Bake until the egg and yolks are cooked through and a very slight jiggle remains, 13 to 15 minutes. Run a butter knife or rubber spatula around each egg to release it. Set aside.
In a small bowl, mix together the mayonnaise, ketchup, relish, and remaining 1/2 teaspoon each of Worcestershire sauce and onion powder. Stir to combine, then season to taste with salt and pepper; set aside.
Line a small rimmed baking sheet with foil. Heat the oil in a medium skillet over medium-high heat. Add the patties and cook until browned but slightly pink in the center, about 2 minutes per side. Transfer the patties to the prepared baking sheet pan.
Place an oven rack about 6 inches from the broiler. Preheat the broiler. Top each patty with 1 slice of cheese. Broil until the cheese is just melting, 30 to 60 seconds.
To serve, top each baked egg with a cheesy patty followed by 3 avocado slices and a drizzle of sauce.