Heat the olive oil in a large heavy pot or Dutch oven over medium heat until shimmering.
Add the chopped onion, carrots, and celery and cook, stirring occasionally, until the vegetables are softened and beginning to color, about 6–8 minutes.
Stir in the garlic and cook until fragrant, about 30 seconds. Push the vegetables to the side of the pot.
Increase heat to medium-high and add the ground turkey. Break it up with a wooden spoon and let it brown without stirring for a minute or two to build flavor, then continue breaking it up until no pink remains, about 5–7 minutes.
Season the meat and vegetable mixture with salt, pepper and Italian seasoning. Stir in the tomato paste and cook 1–2 minutes, pressing it into the bottom of the pot so it caramelizes slightly.
Pour in the vegetable broth to deglaze the pan, scraping up any browned bits. Add the whole tomatoes (crush them with your hands or a spoon into the pot), and add a small pinch of cinnamon or nutmeg. Stir to combine.
Bring the sauce to a gentle boil, then reduce heat to low and simmer uncovered for 20–25 minutes, stirring occasionally, until the sauce thickens and flavors meld.
While the sauce simmers, cook the tagliatelle in a large pot of generously salted boiling water according to package directions until al dente. Reserve 1 cup of the pasta cooking water before draining.
Taste the sauce and adjust seasoning. If it seems too tight, loosen with a few tablespoons of the reserved pasta water until you reach your desired consistency. Stir in the balsamic vinegar to brighten the tomato flavors.
You can either toss the drained pasta directly into the sauce or plate portions of pasta and spoon the bolognese over the top. Finish each serving with a sprinkle of grated Parmesan and a few fresh thyme or oregano leaves. Let sit 3–5 minutes so the sauce settles before serving.