For the homemade pasta, tip the flour onto a clean work surface and make a well in the centre. Crack the eggs into the well, then gradually draw the flour into the eggs until the mixture comes together as a sticky dough. Knead the dough on a lightly floured work surface until all of the flour has been incorporated and the dough is smooth. Alternatively, blend the flour and eggs in a food processor until the mixture forms small crumbs. Tip it out onto a clean, lightly floured work surface and squish into a ball, then knead for a few minutes until smooth.
Wrap the pasta dough in cling film and chill for 20 minutes in the fridge.
Dust the rollers of your pasta-rolling machine with flour.
Cut the chilled pasta dough into 3 equally sized pieces and flatten them with the palms of your hand to a thickness of 1cm/Â½in.
Starting at the lowest (thickest) setting, feed the dough through the machine, turning the handle with one hand and holding the dough as it comes through the machine with the other.
Change the setting on the pasta machine to the next-thickest setting, flour it again and feed the pasta sheet through the machine again, as before.
Repeat this process 3 or 4 more times, flouring the machine and changing the setting down each time until the last but one setting. You should end up with a long sheet of pasta.
Repeat the rolling process with the remaining pieces of pasta dough.
Fold the rolled pasta sheets over into a manageable size, then cut them lengthways to form strips of linguine about 3mm thick. Hang the linguine over a rack so that it can dry out slightly while you make the ragu.
For the ragu, heat the oil in a large flameproof casserole over a medium heat. Add the onions and garlic and fry for 2-3 minutes, or until softened but not coloured.
Add the carrot and celery and fry for another 2-3 minutes. Add the bacon and fry for a further 1-2 minutes.
Add the beef mince and fry for 2-3 minutes on each side, or until just browned.
Stir in the tomato purÃ©e and continue to cook for 1-2 minutes, then pour in the tinned tomatoes and red wine and bring the mixture to the boil.
Stir in half of the basil leaves, then reduce the heat until the mixture is simmering. Simmer for 45 minutes, or until the sauce has thickened. Season, to taste, with salt and freshly ground black pepper. Stir in the remaining basil leaves just before serving.
To serve, bring a large pan of salted water to the boil. Add the pasta and cook for 3-4 minutes, or until tender. Drain well, then add the pasta to the pan of ragu and mix well to coat the pasta in the sauce. Serve sprinkled with the Parmesan.