For the ragù, heat 3 tablespoons of olive oil in a pan, add the onion, carrot and rosemary, and cook for 3 minutes to soften.Add the mince and stir to brown the beef evenly. Add the tomato purée to a small cup of water and mix, then tip into the pan. Cook for 5 minutes, add another ½ a cup of water and season with sea salt and black pepper. Reduce the heat to low and cook gently for at least 35 minutes, until the ragù is soft and tender. Stir occasionally and add a little extra water if needed to stop the sauce from drying out.For the pasta, sieve the flour into a large bowl or heap on a clean work surface. Make a well in the centre and break in the eggs. With a fork or with your hands, gradually mix the flour and eggs together to form a soft dough, then knead for 5 minutes, or until smooth. Shape the dough into a ball, cover with a clean, damp cloth and leave to rest in the fridge for at least 30 minutes.Dust a clean surface with a little extra flour and roll out the dough until wafer thin. Dust it with flour, then loosely roll it in on itself from the sides. Use a knife to cut it crosswise into long strips, then toss it with your hands to shake off the excess flour.Cook the pasta in a large pan of boiling salted water for 2 to 3 minutes, or until al dente. Drain, reserving a little of the water.Tip the pasta into the ragù and stir until well combined. If necessary, loosen with a little reserved pasta water.To serve, spoon the pasta into bowls and serve with an extra drizzle of olive oil.