In a large pot, heat olive oil and butter over medium heat. Add chopped onion and cook until translucent, about 5 minutes. Stir in minced garlic and dried thyme, cooking for an additional minute until fragrant.

Add sliced mushrooms, cooking uncovered for 8-10 minutes until browned and most moisture has evaporated.

Sprinkle all-purpose flour over the cooked mushrooms and stir to combine. Cook for about 1 minute, stirring continuously.

Gradually pour in chicken broth, stirring constantly. Bring to a gentle simmer and let it cook for 10 minutes.

Stir in shredded rotisserie chicken and half-and-half. Reduce heat to low, heating for about 5 minutes, careful not to boil.

Season with salt and pepper to taste. If desired, stir in chopped fresh parsley before serving.