Heat 1 tablespoon oil in a Dutch oven over MEDIUM-HIGH heat, add the chicken, season with ½ teaspoon salt, and cook, stirring to break it up, until crumbled and cooked through. Transfer to a plate and set aside. (chicken ends up a little chunker than beef and that’s fine).
Add the remaining oil to the empty pan and set it over MEDIUM heat. Add the carrot, onion, celery, and parsley, and cook, stirring frequently, until the vegetables are softened, about 3 to 4 minutes. Stir in the garlic and cook until fragrant, about 30 seconds.
Return the cooked meat to the pan, add the tomato paste, fennel seed, oregano, thyme, and pepper, to taste. Cook, stirring, until the tomato paste and spices are well incorporated. Add the wine and beef or chicken broth and increase the heat to bring it to a simmer. Once bubbling, reduce the heat to LOW and allow the mixture to simmer for about 5 minutes to reduce the wine a bit.
Add the tomatoes and milk and bring to a low boil over MEDIUM-HIGH heat, then reduce to MEDIUM-LOW and simmer, uncovered, for 15 to 20 minutes, until slightly thickened. The tomatoes will soften as they cook, so stir frequently, smashing them against the side of the pan with the spoon to help them break down.
Taste and season with additional salt and pepper, if desired.