Heat the large skillet over medium heat and add the olive oil.
Season the chicken breasts with kosher salt and black pepper, then add to the hot skillet and brown about 5 minutes per side until golden; remove the chicken and set aside.
Add the chicken broth to the skillet and bring to a gentle simmer, scraping up any browned bits from the bottom.
In a mixing bowl, whisk together the room-temperature heavy cream and all-purpose flour until smooth.
Slowly pour the cream-and-flour mixture into the simmering broth while whisking continuously; continue whisking until the sauce begins to thicken, about 5 minutes.
Stir in the dried oregano, dried basil, dried rosemary, red pepper flakes, and half of the grated Parmesan until combined.
Return the browned chicken to the skillet and add the sun-dried tomatoes, then sprinkle the remaining Parmesan over the top.
Let everything simmer together for about 5 minutes more to heat through and finish the sauce, then remove from heat and serve.