Slice chicken breasts horizontally into thin cutlets. Pat dry with paper towels and season generously on both sides with kosher salt and black pepper.
Dredge each chicken cutlet lightly in flour, shaking off any excess.
Heat olive oil and 1 tablespoon butter in a large heavy skillet (cast-iron preferred) over medium-high heat until shimmering. Sear chicken for 4-5 minutes per side until golden brown and cooked through (internal temp 165 F / 74 C). Transfer chicken to a warm plate.
Reduce skillet heat to medium. Add remaining tablespoon of butter and minced garlic; saute for 60 seconds until fragrant.
Pour in low-sodium chicken broth, scraping up all the golden browned bits from the bottom of the pan with a wooden spoon. Simmer for 2 minutes until reduced slightly.
Stir in heavy cream, grated Parmesan cheese, julienned sun-dried tomatoes, dried oregano, dried thyme, and crushed red pepper flakes. Simmer gently for 3-4 minutes until the sauce thickens into a rich, silky glaze.
Return seared chicken cutlets and any accumulated juices back into the skillet. Spoon the velvety sauce over the chicken and simmer for 2 minutes to heat through.
Remove from heat, garnish generously with fresh basil ribbons and extra grated Parmesan, and serve hot over pasta, steamed rice, or with warm garlic bread.