In a food processor, pulse the chard until finely chopped. Transfer to a medium bowl, rinse out the food processor and return it to the base.
In a soup pot , heat 3 tbsp. olive oil, three turns of the pan, over medium-high. Add the carrots, chopped onion, celery, rosemary, chopped garlic and chile; season with salt and pepper. Cover, reduce the heat to medium and cook, stirring occasionally, until the vegetables are tender but not brown, 7 to 8 minutes.
Stir in the stock, chickpeas and tomatoes. Transfer half the soup to the food processor and coarsely puree. Return to the pot and stir in the chard; season. Simmer, stirring often, until the flavors blend, about 5 minutes.
Preheat the broiler. Toast the bread under the broiler until charred, about 2 to 3 minutes per side. Rub the toasts with the halved garlic clove, drizzle with EVOO and sprinkle with sea salt. Place 1 toast in each soup bowl and top with soup. Garnish with the raw onion and cheese. Pass more EVOO at the table for drizzling.