Preheat the oven to 450°F or 230°C.
Remove most of the outer peel of the garlic and cut off the top part of the head of the garlic. Drizzle the exposed garlic with the olive oil. Wrap in foil and place on a parchment lined baking sheet. Cut the onion into wedges and oil with the olive oil.
Lightly oil the peppers with vegetable oil. Place everything on the baking sheet. Roast everything for about 30/40 minutes rotating half way until the peppers have a good amount of char marks. Once done, place the hot peppers in a bowl and cover with cling film or aluminium foil. Let them rest until cool. Gently remove the skins and the seeds from the inside of the pepper. Squeeze out the garlic.
In a pot, warm up the vegetable stock with the tomato paste.
When everything is ready, add the roasted vegetables to the stock with the basil leaves. Blend everything, either with an immersion blender or a normal one, until everything is nice, smooth and velvety. Season the soup to your liking.
Bring the soup to light simmer. Serve with croutons on top, some fresh basil leaves and a generous drizzle of olive oil.