Slice each of the 4 (9- to 10-oz.) boneless, skinless chicken breasts in half horizontally to create 8 thin, even cutlets. Season both sides of the chicken with Kosher salt and Ground black pepper, to taste.

Spread the 1/2 cup all-purpose flour in a shallow dish. Dredge each chicken cutlet, one at a time, in the flour, gently shaking off any excess. Transfer the floured cutlets to a plate.

Heat a large skillet over medium-high heat. Add 2 Tbsp. salted butter and 1 Tbsp. vegetable oil. Once the butter has melted and the foam has subsided, add the first 4 floured chicken cutlets to the pan. Cook until golden brown, about 4 to 5 minutes per side. Transfer the cooked chicken to a clean plate.

Repeat the searing process with another 2 Tbsp. salted butter, 1 Tbsp. vegetable oil, and the remaining 4 chicken cutlets. Once cooked, transfer them to the plate with the first batch. Reduce the heat to medium-low.

Melt the remaining 2 Tbsp. salted butter in the same skillet. Add the 1/2 small yellow onion, finely diced, and cook until translucent, about 2 minutes. Stir in the 4 garlic cloves (grated or minced), 1/4 tsp. crushed red pepper, and 2 Tbsp. tomato paste. Cook, stirring frequently, until the garlic is fragrant and the tomato paste darkens slightly, about 1 minute.

Pour in the 1/2 cup dry white wine to deglaze the pan, scraping up any browned bits from the bottom. Simmer the liquid until the wine has reduced by about half, approximately 2 minutes.

Stir in the 2 cups chicken broth, 2/3 cup dry-packed sun-dried tomatoes (sliced), and 1/2 tsp. Italian seasoning. Bring the sauce to a gentle simmer. Return the seared chicken and any accumulated juices from the plate to the skillet. Cook for 5 minutes, allowing the sauce to thicken slightly and the chicken to warm through.

Stir in the 1 cup heavy cream and 1/2 cup freshly grated parmesan. Continue to simmer for 5 minutes more to ensure the chicken is tender and all the flavors are well-melded.

Top with finely chopped Flat leaf parsley and extra freshly grated parmesan, if desired. Serve immediately.