Brush the burgers on both sides with 2 tablespoons of olive oil and season well with salt and pepper and place on a hot grill (450°) and cook. Cook for 4 to 6 minutes on each side to achieve a medium internal temperature. With 2 minutes remaining in the cooking process, add on the Gruyere cheese to melt. Note: the burger, mushrooms, and onions are all placed on the grill at the same time.
Rub the Portobello mushrooms with 3 tablespoons of olive oil and season well with salt and pepper and place them cap side up on a hot grill (450°) for 2 minutes. Next, flip them over and place each one on a small individual piece of aluminum foil and fold up the sides to mold around the outside of the Portobello and crack an egg into each Portobello mushroom. Season the egg with salt and pepper and cook until the egg is to the desired doneness (about 8 to 10 minutes for over-medium).
Brush the sliced onions with the remaining olive oil, season with salt and pepper, and place on a hot grill (450°) and cook on each side until caramelized (about 4 to 6 minutes on each side)
Toast the buns on the grill until golden brown.
Serve the cheeseburger on the bottom bun with the onion and egg Portobello on top.
Optional French Remoulade Sauce: Add all ingredients into a small bowl and whisk together until completely combined. Keep refrigerated before serving along side the burger.