In a large saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until softened and light golden-brown, 8 to 10 minutes. Add the thyme and carrot and cook until the carrot is tender, about 5 minutes.
Using your hands, crush the tomatoes and add them with their juice to the saucepan. Bring to a boil, stirring, then lower the heat and simmer, stirring occasionally, until thickened, about 30 minutes. Season to taste with salt.