Warm a large heavy skillet or Dutch oven over medium heat. Add the olive oil, then sauté the chopped onion, grated carrot, and diced celery until softened and translucent, about 6 to 7 minutes, stirring occasionally so they don't brown.
Push the vegetables to one side, increase the heat to medium-high and add the ground turkey in an even layer. Break it up with a wooden spoon and let it brown without stirring for 2 minutes, then continue cooking and stirring until no pink remains, about 4 more minutes.
Stir the garlic into the meat-vegetable mixture and cook 30 seconds until fragrant. Add the tomato paste, mixing it into the meat and vegetables; cook 1 minute to develop its flavor.
Pour in the crushed tomatoes and chicken broth, then sprinkle in the dried oregano, dried basil, salt, and black pepper. Bring the sauce to a low simmer, then reduce heat to maintain a gentle simmer. Partially cover and cook, stirring occasionally, for 25 to 30 minutes so the sauce thickens and flavors meld.
Stir in the milk and simmer uncovered for another 3 to 5 minutes to round out acidity and add silkiness to the sauce. Taste and adjust seasoning if needed.
If using pasta, toss the cooked pasta with a few ladlefuls of sauce in the pot to coat, or plate pasta and spoon the Bolognese over the top. Serve with grated Parmesan and torn basil if desired. Let the sauce rest for 3 to 5 minutes before serving for best texture.