In a 10 to 12 inch sauté pan, heat the olive ovil over medium-high heat until hot. Add the garlic, increase the heat to high, and cook, stirring, until golden, about 2 minutes. Add the olives and cook for 3 minutes, or until softened.
Add the tomato sauce, lower the heat, and simmer for 15 to 20 minutes, occasionally basting the olives with the tomato sauce. Serve warm or at room temperature.
In a 3 quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, 8 to 10 minutes. Add the thyme and carrot and cook until the carrot is quite soft, about 5 minutes.
Add the tomatoes, with their juice, and bring to a boil, stirring often. Lower the heat and simmer until as thick as hot cereal, about 30 minutes. Season with salt. The sauce can be refrigerated for up to 1 week or frozen for 6 months.