Preheat oven to 425 degrees F.
Toss the tomatoes in 1 tablespoon olive oil, salt, and pepper. Cut the eggplant in half lengthwise and score the flesh in a crisscross pattern with a sharp knife. Drizzle 1 tablespoon oil on the eggplant and season with salt and pepper. Lay the tomatoes, eggplant, whole garlic cloves, and bell pepper on a sheet pan. Roast for 20-25 minutes or until the vegetables are charred and softened.
Transfer the bell pepper to a bowl and cover with plastic wrap. Let sit for 10 minutes. Remove the cover and peel the skin off the bell pepper. Discard the stem and seeds.
Peel off the garlic skin and scoop out the eggplant flesh. Place the eggplant flesh, garlic, tomatoes, and bell pepper in a blender. Add the remaining 2 tablespoon olive oil, 2 Calabrian chilies plus 2 tablespoon chile oil, olives, red wine vinegar, and basil and blend until smooth. Season the sauce with salt and pepper.
Peel and thinly slice one medium onion. Heat 2 tablespoon olive oil in a large saute pan over high heat and add the onions. Season them with salt and continue to cook until the onions are softened and translucent about 7-8 minutes. Reduce the heat to medium-low and continue to cook until the onions are a deep brown and caramelized, stirring occasionally. If the pan becomes too dry add 1-2 tablespoon water. Remove from heat and set aside.
Slice 4 ciabatta buns in half. Spread the spicy roasted vegetable sauce on the bottom bun and place a slice of cheddar on top. Place the buns on a baking sheet and heat them in the oven for 2-3 minutes or until the cheese is melted. Remove from the oven and top the cheese with a fried egg, caramelized onions, and fresh arugula. Add extra sauce if desired and top with the remaining bun. Serve immediately.