Peel onion and thinly slice into rings. Add oil to a large skillet and add onions. Season with salt and pepper. Cook on medium heat, stirring often. When pan is dry, add a splash of wine every so often. Cook onions for about 25-30 min or until onions have reduced drastically in size and become browned and caramelized.
Preheat oven to broil. Slice bread and add slice of cheese to each piece. Broil for about 3-4 min or until browned and melty.
Meanwhile, melt butter in a skillet. Whisk eggs and when skillet is very hot, pour eggs into pan. Add salt, pepper and a few more pieces of cheese. This will make the eggs super creamy and extra fluffy. Using a spatula, fold the eggs toward the middle of the pan. Turn heat off after 1 min as the eggs will continue to cook.
Season some arugula with lemon juice, salt and pepper. Add eggs to melty cheese bread, and add onions and arugula. Enjoy!