In a large Dutch oven or heavy-bottomed pot, heat the olive oil and butter over medium heat. Add the chopped onion, carrots, and celery. Cook, stirring occasionally, until softened, about 5-7 minutes. Add the minced garlic and dried thyme, and cook for one more minute until fragrant.

Add the chicken pieces to the pot, season with salt and pepper, and cook until lightly browned on all sides and cooked through, about 5-6 minutes.

Sprinkle the flour over the contents of the pot and stir constantly for 1 minute. Add the uncooked orzo and stir for another minute to lightly toast it.

Gradually pour in the warm chicken broth while stirring and scraping the bottom of the pot to release any browned bits. Bring to a gentle simmer, then reduce heat to low, cover the pot, and let it cook for 10-12 minutes, or until the orzo is al dente. Stir occasionally to prevent sticking.

Remove the pot from the heat. Stir in the heavy cream, frozen peas, and fresh parsley. The heat from the dish will warm the peas through. Let it sit for 2-3 minutes for the sauce to thicken.

Taste and adjust seasoning with salt and pepper if needed. Serve hot.