Place onion, olive oil, and chopped garlic in a medium-large saucepan over medium heat, stirring occasionally for 15 minutes, or until onions are very soft, translucent and caramelized.
Add balsamic, maple syrup or sugar and slivered garlic and continue to cook and stir for 2 to 3 minutes.
Add tomatoes with their juices and bread pieces. Season, to taste, with salt and pepper. Add vegetable broth or water and bring to a simmer. Allow to stand until bread absorbs liquid. Add more vegetable broth or water if the soup becomes too thick. Remove the soup from the heat and cool a touch.
Tear basil leaves and stir into soup with additional olive oil. Allow to cool to serving temperature to let the bread absorb the flavors of the basil and olive oil.
If desired, drizzle each serving with olive oil, a pinch of coarse sea salt and freshly ground black pepper.
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