Slice each chicken breast in half horizontally to make two thinner cutlets (you should have four pieces total). Pound lightly if needed for even thickness.
Combine 2 teaspoons Italian seasoning with the salt and pepper. Season one side of each chicken piece with this mixture.
Heat about 2 tablespoons of the oil from the sun-dried tomato jar in a large skillet over medium-high heat. When hot, place the chicken seasoned-side down in the pan.
Season the top side of the chicken with the remaining seasoning, then cook undisturbed about 4 minutes. Flip and cook another 3–4 minutes until golden and cooked through (165°F internal).
Remove the chicken to a plate and cover loosely with foil to keep warm.
Reduce heat to medium and add a small additional drizzle (~1 teaspoon) of the sun-dried tomato oil to the skillet, then add the minced garlic. Cook about 10 seconds, stirring, until fragrant but not browned.
Add 1/2 cup chicken broth and scrape up any browned bits from the pan. Let it simmer until mostly reduced, about 1 minute.
Add the remaining 1/2 cup chicken broth, heavy cream, sun-dried tomatoes, red pepper flakes (if using), the remaining 1 teaspoon Italian seasoning, Dijon mustard, and grated Parmesan. Stir and simmer until the cheese melts and the sauce thickens, about 2–4 minutes. Taste and adjust seasoning.
Stir in the ribboned basil, return the chicken (and any accumulated juices) to the pan, and spoon the sauce over the chicken. Serve immediately.