In a large nonstick pan heat olive oil over medium heat. Add carrots, celery, and garlic. Season with salt and sauté until fragrant, about 5 minutes.
Add beef, stirring to break into small pieces as it browns. When meat is no longer pink, stir in tomato paste to coat the mixture.
Deglaze the pan with wine, stirring to remove any stuck-on bits from the pan. Stir in milk until incorporated. Add tomato puree and Italian seasoning. Add another pinch of salt, stir, and cover until bubbling. Stir and reduce heat to low. Cover and simmer for 1 hour.
Stir in chicken broth, increase heat to return to a boil. Stir in Orzo to cook pasta in sauce. Stir consistently to prevent pasta from sticking to the bottom of the pot. Pasta should cook in 10 minutes. If the sauce is too thick, add ½ cup of water and stir to combine.
Once pasta is cooked, add cheese, and stir. Season with salt and pepper to taste. Serve immediately. Garnish with more freshly grated cheese, if desired.
