Heat 2 tablespoons olive oil in a large non-stick saucepan on medium heat. Add diced onions and carrot and saute, uncovered, for 5-7 minutes until onions are soft and liquid is released, stirring occasionally.
Cover the saucepan and saute for a further 20-30 minutes, being careful not to burn the onions. If the mixture starts to dry out, add 2-4 tablespoons of water.
In a separate large skillet, heat 2 tablespoons olive oil on high heat. Add ground beef and cook until browned. Add ground meat to saucepan with onion and carrot mixture.
In the same skillet used for cooking the meat, add wine on high heat and use a spatula to scrape the pan. Once alcohol evaporates (2-3 minutes) and mixture is bubbly, carefully pour wine mixture into saucepan with meat and onions.
Add nutmeg and bayleaf to meat mixture. Add canned tomatoes, tomato paste (if using), and 2-4 tablespoons of water or milk (for a more creamy sauce). Season sauce with salt and pepper, to taste. Simmer, uncovered, on medium-low heat for a further 20-30 minutes.
In the meantime, cook pasta in salted water according to package instructions (or until al dente). Drain and divide pasta between 4 serving plates.
Top with bolognese sauce, chopped fresh parsley, and freshly grated Parmesan cheese and serve warm.