Boil chicken broth for rice: Preheat oven to 350°F. Add 2 cups of the chicken broth and ½ teaspoon of the salt to a medium saucepan. Bring to a boil over medium-high; stir in rice, and return to a boil.

Cook rice: Reduce heat to medium-low; cover and cook until rice is just beginning to soften and broth is mostly absorbed, about 15 minutes.

Melt butter and oil: Meanwhile, heat oil and 2 tablespoons of the butter in a large Dutch oven over medium-high.

Season chicken: As butter melts, season chicken all over with black pepper and remaining 1 teaspoon salt.

Cook chicken: Add chicken to Dutch oven, and cook, flipping once, until chicken is golden brown and almost cooked through, about 6 minutes per side.

Remove chicken from pan: Transfer chicken to a cutting board to rest. Do not wipe Dutch oven clean.

Add butter to pan drippings: Reduce heat to medium, and add remaining 2 tablespoons butter to drippings in Dutch oven.

Cook sun-dried tomatoes, herbs, and spices: Once butter is melted, stir in sun-dried tomatoes, shallot, garlic, oregano, and crushed red pepper; cook, stirring often, until shallot softens, about 3 minutes.

Add tomato paste: Add tomato paste; cook, stirring constantly, until tomato paste coats vegetables, about 1 minute.

Add wine and broth to pan: Stir in wine and remaining ½ cup chicken broth; cook, scraping any flavorful bits from the bottom of the Dutch oven, until reduced by one-quarter, about 3 minutes.

Add rice mixture to broth mixture, along with remaining ingredients: Stir in rice mixture (including any unabsorbed broth), heavy cream, balsamic vinegar, and ¼ cup of the Parmigiano-Reggiano; cook, stirring often, until Parmigiano-Reggiano melts into the sauce, about 1 minute. Remove from heat.

Add rice and chicken to baking dish: Transfer rice mixture to a (3-quart) baking dish. Nestle chicken breasts into rice mixture.

Bake casserole: Bake in preheated oven until flavors meld and chicken is cooked through, about 30 minutes. Top evenly with remaining 2 tablespoons Parmigiano-Reggiano, and sprinkle with fresh basil.