Heat 1 tablespoon olive oil in a 12-inch skillet over medium-high heat. While the oil heats, pat the chicken breasts dry and season both sides with salt and ground black pepper to taste.
Add the chicken to the hot skillet and cook 3–5 minutes per side, until browned and no longer pink in the center (internal temperature should reach 165°F). Remove the chicken from the skillet and set aside on a plate.
Reduce heat to medium and add the remaining 1 tablespoon olive oil to the skillet. Add the diced onion and minced garlic and cook, stirring, about 2 minutes, until the onion is tender/translucent.
Add the tomato sauce, heavy whipping cream, 1/4 cup grated parmesan cheese, and 1 teaspoon Italian seasoning to the skillet. Stir to combine and bring the sauce to a gentle simmer over medium heat. Taste the sauce and season with additional salt and pepper if needed.
Return the cooked chicken to the skillet, spooning some sauce over each piece. Evenly sprinkle the 1/2 cup shredded parmesan cheese over the chicken.
Cover the skillet with a lid and cook 1–2 minutes, just until the shredded parmesan has melted. Remove from heat.
Garnish with additional Italian seasoning if desired and serve warm.