Put the olive oil and garlic in a medium pot, turn the heat to low and cook the garlic, stirring occasionally at first and more often as it colors, until golden brown, about 7 minutes. Raise the heat to medium-low, add the onions and a large pinch of salt and cook, stirring occasionally, until the onions have softened, about 5 minutes. Add the carrots, and cook, stirring occasionally, until softened, about 5 minutes. Add the tomato paste and the red pepper flakes, and cook, stirring constantly, for 2 minutes.
Add the tomatoes, bay leaves, thyme, basil sprig and 1 teaspoon salt, raise the heat to medium and bring the sauce to a lively simmer. Cook, stirring frequently, until quite thick (if you draw a spoon through the sauce, you'll see the bottom of the pot for a few seconds before the sauce fills back in), about 30 minutes. As the sauce thickens, turn the heat down a bit so that the bottom doesn't scorch. Stir in the chopped basil. (The sauce can be cooled to room temperature, then refrigerated for up to 3 days.)
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