Preheat the oven to 375 degrees. Make beef patties. Mix beef, salt, pepper, onion powder, and dried onions until just blended. Divide meat into six 6-ounce portions and place on a baking sheet. Lightly flatten into patties 3/4-inch thick. Press the center of the patties down with a small glass creating a well for the egg in each patty. Add eggs to the burgers and cook in the oven for 25-30 minutes, or until the eggs are thoroughly cooked.
To make the caramelized onions, melt butter in a large skillet over medium-high heat. Add onions and cook until tender, about 8 minutes. Season with salt and pepper.
To make the sauce, mix together the mayonnaise and barbecue sauce.
To assemble, spread the sauce on the bottom buns and the honey dijon mustard on the top buns. Place the lettuce, tomato, and patty on the bottom bun. Top with caramelized onions and the top bun.
