Warm a skillet over medium-high heat and lightly coat the chicken breasts with the oil; place them in the hot pan, letting them sear without oil in the pan. Cook chicken for about 3 to 4 minutes per side until nicely browned and it releases naturally from the pan. Remove the chicken and set aside, knowing it will finish cooking with the rice. Toss the sliced mushrooms into the same hot pan and cook for 4 to 5 minutes until they are nicely caramelized and golden brown, skipping the salt to achieve perfect color. Stir in the uncooked rice, chopped garlic, and dried oregano; sauté the mixture for 2 to 3 minutes until the garlic releases its aroma. Pour in the chicken stock, add the cream, salt, and pepper, stirring gently to combine. Nestle the chicken breasts on top of the rice mixture and lower the heat to medium. Cover the pan with a lid and allow everything to simmer for 20 to 25 minutes until the liquid has been absorbed and the chicken is thoroughly cooked, reaching an internal temperature of 165°F. Sprinkle with sliced green onions if desired and serve alongside steamed vegetables like broccoli or cauliflower for a complete meal.