Pat the chicken dry and season both sides with salt, pepper, and dried thyme. Heat olive oil in a large skillet over medium-high heat. Add the chicken and cook until golden brown, about 4-5 minutes per side. Transfer to a plate and set aside.
In the same skillet, melt the butter over medium heat. Add the mushrooms and onion and cook, stirring occasionally, until softened and lightly browned, about 5-6 minutes. Add the garlic and cook for 30 seconds until fragrant.
Stir in the wild rice blend and cook for 1 minute, then pour in the chicken broth. Bring to a simmer, then reduce the heat to low, cover, and cook for 25 minutes.
Return the chicken to the skillet, nestling it into the rice. Cover and cook for an additional 10-15 minutes, or until the rice is tender and the chicken is cooked through (internal temperature reaches 165°F).
Stir in the heavy cream and Parmesan cheese. Cook uncovered for 2-3 minutes, until the sauce thickens slightly. Taste and adjust seasoning with salt and pepper.
Garnish with fresh parsley, if desired, and serve hot.