Make a mound of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs, oil and any other flavoring you choose. Using a fork, beat together the eggs, oil and flavorings and begin to incorporate the flour starting with the inner rim of the well. As you expand the well, keep pushing the flour up to retain the well shape. Do not worry that this initial phase looks messy. The dough will come together when 1/2 of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands primarily. Once you have a cohesive mass, remove the dough from the board and scrape up any left over crusty bits. Lightly flour the board and continue kneading for 3 more minutes. The dough should be elastic and a little sticky. Continue to knead for another 3 minutes, remembering to dust your board when necessary. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature. Note: do not skip the kneading or resting portion of this recipe, they are essential for a light pasta.
Saute the onion and garlic in the olive oil over medium heat until translucent, but not brown (about 10 minutes). Add the thyme and carrot and cook 5 minutes more. Add the tomatoes. Bring to a boil, lower the heat to just bubbling, stirring occasionally for 30 minutes. Season with salt to taste. Serve immediately, or set aside for further use. The sauce may be refrigerated for up to one week or frozen for up to 6 months.