Heat 1 1/2 tablespoons of the olive oil in a 12-inch skillet over medium-high heat.
Add the chopped onion, sliced carrots, and sliced celery and sauté until tender, about 8–9 minutes; transfer the vegetables to a plate.
Add the remaining 1/2 tablespoon olive oil to the same skillet and return to medium-high heat.
Add the chicken pieces and cook about 3 minutes, then add the minced garlic and cook about 2 more minutes until the chicken is nearly cooked through.
Stir in the rice, chicken broth, and minced thyme/rosemary; season with salt and pepper to taste. Return the sautéed vegetables to the skillet and bring the mixture to a boil.
Reduce heat to medium-low, cover, and simmer until most of the liquid is absorbed and the rice is nearly tender, about 18–20 minutes.
Stir in the frozen peas and heavy cream, cover again, and cook until the rice is fully tender and heated through, about 5 more minutes.
Remove from heat, stir in the Parmesan and minced parsley, adjust seasoning if needed, and serve warm.