Preheat the oven to 350 degrees F (175 degrees C). Sprinkle chicken with 1/4 teaspoon salt and 1/4 teaspoon pepper.
Heat 1 tablespoon olive oil in an oven-safe 12-inch skillet over medium-high heat. Add chicken and cook until browned, 2 to 3 minutes per side (chicken will not be done at this point). Remove chicken from skillet.
Add remaining 1 tablespoon olive oil and butter to skillet. When hot, add mushrooms, remaining 1/2 teaspoon salt and 1/4 teaspoon pepper to skillet. Cook and stir until mushrooms are tender and liquid has evaporated, 3 minutes.
Add shallot, garlic, thyme, and crushed red pepper to skillet. Cook and stir until tender and aromatic, 1 to 2 minutes.
Add wine and cream to skillet. Stir with a wooden spoon to scrape up any browned bits from the bottom of the skillet. Bring to boiling. Reduce heat to medium and simmer until liquid has reduced by half, 5 minutes.
Stir in rice. Add broth to the skillet. Return to boiling over high heat. Remove skillet from heat. Return chicken to the skillet and cover with a lid.
Bake in the preheated oven until rice is tender, 45 to 50 minutes. Slice chicken and return to skillet. Garnish with parsley.
