Pat the Chicken thighs and drumsticks dry with paper towels and season generously with Fine sea salt and Black pepper.
Heat a few tablespoons of Extra-virgin olive oil in a large heavy skillet or Dutch oven over medium-high heat. Working in batches so as not to crowd the pan, brown the chicken well on all sides, about 6–8 minutes per side; transfer the browned pieces to a plate and set aside.
Reduce the heat to medium and add the chopped Onion, Carrot and Celery to the pot. Cook, stirring occasionally, until the vegetables are softened and beginning to caramelize, about 8–10 minutes.
Add minced Garlic and cook for 30–60 seconds until fragrant.
Pour in the Red wine to deglaze the pan, scraping up any browned bits from the bottom. Let the wine simmer and reduce by about half, 3–5 minutes.
Stir in the Canned San Marzano Tomatoes, crushing them with a spoon if whole, then add a sprig or two of Rosemary, a few torn Sage leaves, and Bay leaves. Add a splash of Red wine vinegar to brighten the sauce and season with a little more salt and pepper.
Return the browned chicken to the pot, nestling the pieces into the sauce. Bring to a gentle simmer, cover partially, and cook over low heat until the chicken is tender and cooked through, about 30–40 minutes.
If you prefer a thicker, more concentrated sauce, remove the lid for the last 10–15 minutes to reduce. Taste and adjust seasoning with more salt, pepper, or a final splash of vinegar if needed.
Serve the Pollo alla Cacciatora hot, spooning plenty of sauce over the chicken and alongside polenta, pasta, or crusty bread to soak up the juices.
