Heat olive oil in a large pot over medium heat. Add the carrot, celery, and onion and cook for 8-10 minutes, stirring occasionally, until softened and lightly golden.
Add the ground beef and garlic, breaking the meat up as it cooks, for 6-8 minutes until browned.
Pour in the red wine and simmer for 3-4 minutes, scraping up any browned bits, until mostly evaporated.
Add the crushed tomatoes, bay leaf, and 1 tsp salt. Bring to a gentle simmer, then reduce heat to low and cook uncovered for 45 minutes to 1 hour, stirring occasionally, until deeply thickened.
Stir in the milk during the last 10 minutes of cooking to mellow the acidity. Remove the bay leaf and season to taste.
Meanwhile, cook the tagliatelle in well-salted boiling water until al dente. Reserve ½ cup pasta water, then drain.
Toss the pasta with the bolognese, loosening with a splash of pasta water if needed. Serve topped with grated parmesan.
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