Season the chicken. Pat the chicken dry and season both sides with salt, pepper, and a pinch of smoked paprika if using.
Brown the chicken. Heat olive oil in a large, deep skillet or Dutch oven over medium-high heat.
Add the chicken and sear 3–4 minutes per side until lightly golden. Transfer to a plate; it won’t be cooked through yet.
Soften the aromatics. Lower the heat to medium. Add a drizzle more oil if the pan looks dry.
Add the onion and a pinch of salt. Cook 3–4 minutes, stirring, until softened. Stir in garlic, red pepper flakes, and oregano.
Cook 30–60 seconds until fragrant.
Deglaze and build the sauce. Pour in the chicken broth and use a wooden spoon to scrape up any browned bits. Stir in the crushed tomatoes, balsamic vinegar, and sugar or honey. Bring to a gentle simmer.
Simmer the chicken. Return the chicken and any juices to the pot, nestling the pieces into the sauce.
Cover and simmer on low for 12–15 minutes (thighs may need up to 18 minutes), flipping once, until the chicken is cooked through.
Finish the sauce. If you want it creamy, stir in the heavy cream and Parmesan and simmer uncovered for 2–3 minutes to thicken slightly. Taste and adjust with salt, pepper, or more balsamic as needed.
Add fresh basil. Turn off the heat and stir in most of the basil, saving a little for garnish. The residual heat will release its aroma without dulling the flavor.
Serve. Spoon the chicken and sauce over pasta or rice, or serve with crusty bread.
Top with the remaining basil and extra Parmesan if you like.