Preheat your oven to 350°F. In a pot, bring 2.25 cups chicken broth and 0.75 teaspoon salt to a boil over medium-high heat. Stir in the rice mix, return to a boil, then reduce heat to low and cover. Simmer for 15 minutes until the rice is tender and liquid is absorbed. This gives you a head start on cooking while you prepare the chicken and sauce.

While the rice cooks, season the chicken breasts evenly on both sides with 0.75 teaspoon salt and 0.25 teaspoon black pepper. Heat 2 tablespoons olive oil and 2.5 tablespoons butter in a large skillet over medium-high heat until shimmering. Sear the chicken for 6 minutes per side until golden brown on the outside (it won't be fully cooked through). Transfer the seared chicken to a cutting board and set aside. I like to keep the pan on the heat because you'll use those flavorful browned bits for your sauce.

In the same skillet with the chicken drippings, melt 2.5 tablespoons butter over medium heat. Add the minced shallot and garlic, cooking for 1 minute until fragrant. Stir in the sun-dried tomatoes, dried oregano, and red pepper flakes, cooking for 2 additional minutes to release the flavors. Add the tomato paste and cook for 1 minute, then deglaze the pan with the dry white wine and 0.5 cup chicken broth, scraping up any browned bits from the bottom. Simmer for 3 minutes to reduce slightly and mellow the acidity. This layering of aromatics and tomato paste creates the depth that makes this dish so special.

Remove the sauce from heat and stir in the heavy cream, balsamic vinegar, and 1.75 ounces grated Parmesan cheese until smooth and well combined. Season to taste with a pinch of salt and pepper if needed. Add the cooked rice from Step 1 to the sauce and fold gently to combine everything without breaking up the rice grains too much. This cream-based sauce is what gives the dish its rich, luxurious character.

Spread the rice and sauce mixture into a 9x13 inch baking dish. Nestle the seared chicken breasts from Step 2 into the rice mixture, pressing them down slightly so they're partially submerged. Bake uncovered at 350°F for 30 minutes until the chicken is cooked through (internal temperature of 165°F) and the casserole is heated through.

Remove the casserole from the oven and sprinkle the remaining Parmesan cheese and fresh thyme leaves over the top as garnish. Let it rest for 5 minutes before serving to allow the flavors to settle and the chicken to retain its juices.