Pat the chicken pieces dry and season with salt and pepper. Heat olive oil and butter in a 12-inch cast iron skillet over medium-high heat. Add chicken in a single layer and sear for 3-4 minutes per side, until a golden-brown crust forms. Remove chicken to a plate.

Reduce heat to medium. Add sliced mushrooms to the skillet and cook for 5-7 minutes until browned. Add the shallot and cook for 2 minutes until softened. Stir in the garlic and 1 tbsp of thyme and cook for 30 seconds until fragrant.

Add the dry orzo to the skillet and stir for 1 minute to toast. Pour in the chicken broth, scraping the bottom of the pan to deglaze. Bring to a simmer, then reduce heat to medium-low. Cook for 8-10 minutes, stirring occasionally, until orzo is al dente and most of the liquid is absorbed.

Stir in the heavy cream and Parmesan cheese until melted and the sauce is creamy. Return the seared chicken and any juices back to the skillet. Stir to combine and heat through. Season to taste with more salt and pepper.

Garnish generously with fresh thyme leaves and serve immediately, directly from the skillet.