In the bowl of an electric mixer with the whisk attachment, blend the flour, garlic, parsley, thyme, salt, and pepper. Add the water and mix well on low speed until a dough-like mixture is formed. (The water amount may need to be adjusted so the dough is moist but not wet.) Once mixed, wrap the dough in plastic wrap and place in the refrigerator for 30 minutes.
Preheat the oven to 400 degrees F.
Split the dough in half and roll out on a floured surface to 1/8-inch thickness. Transfer to a baking sheet and place in the oven for 10 minutes. Remove from the oven and evenly spread 1/2 cup Tomato Sauce and half the cheese. Add additional toppings, if desired. Bake for a second time until the cheese is melted, 8 to10 minutes. Repeat with the other dough ball.
In a large saucepan over medium-high heat, heat the oil. Add the onions and garlic and cook until soft. Add the carrots and wine, deglazing the pan. Reduce the heat to medium and add the thyme, cooking for an additional 5 minutes. Add the tomatoes and their juices and bring to a boil. Reduce the heat to a simmer and add the basil. Continue to cook over low heat for 20 to 30 minutes. Taste for seasoning and remove from the heat.