- Preheat an electric grill griddle to high according to the manufacturer’s instructions.
- In a bowl, toss the onion with the 1 tablespoon vegetable oil, and season with salt and pepper. Place on the grill and cook, turning once, until tender, about 10 minutes. Transfer to a bowl.
- Place the bacon on the grill and cook, turning once, until crispy, about 8 minutes. Drain on a paper towel-lined plate.
- Meanwhile, oil the griddle with the 1 teaspoon vegetable oil. Working in batches, cook the eggs until the whites are set and the yolks are still runny, 2 1/2 to 3 minutes. Season with salt and pepper. Transfer to a plate.
- Spread 1 tablespoon mayonnaise on each slice of toast. Top 4 of the slices with the tomato slices, grilled onions, 2 bacon slices, arugula and 1 egg. Cover with the remaining toast, mayonnaise side down. Serve immediately.