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Make the croutons: Preheat the oven to 400°F. In a small saucepan, combine the oil, garlic, and 1/2 teaspoon of the salt, bring the oil to a simmer over medium heat, and let the garlic simmer for 5 minutes, without browning, to infuse flavor. Discard the garlic; reserve the oil.
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Line a rimmed baking sheet with parchment paper. Place the bread cubes on prepared pan, drizzle with the garlic oil, sprinkle with the remaining 1/2 teaspoon salt, and toss to help the bread absorb the oil. Spread the cubes in a single layer in the pan and bake for 5 minutes. Toss the croutons, and sprinkle with half of the Parmesan cheese. Bake until the croutons are golden and the cheese is melted, another 5 minutes. Toss the croutons again, gathering up the cheese onto the croutons, then sprinkle with the remaining Parmesan and bake until golden brown and crispy, another 5 to 8 minutes.
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Caramelize the onions: In your largest Dutch oven or heavy pot, heat the butter over medium-high heat. Add the onions, shallots, garlic, thyme sprigs, bay leaf, 1 teaspoon salt, and ¼ teaspoon pepper and cook, stirring occasionally and scraping up any brown bits at the bottom of the pan, until the onions are very soft and dark golden brown, 40 to 60 minutes, depending on your heat. (Decrease heat to medium if the onions are burning.)
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Add the wine, scrape the brown bits on the sides of the pot into the onions, bring to a boil, lower the heat, then simmer, stirring occasionally, until the onions are nearly dry, about 10 minutes. Remove the thyme sprigs and bay leaf. Add the broth, stir, bring to a light boil, reduce the heat to low, and simmer for 30 minutes, stirring occasionally. Season with 1 to 1½ teaspoons salt, depending on the saltiness of the broth, and pepper to taste. Serve hot with croutons and a sprinkle of fresh thyme leaves.