If the chicken breasts are especially thick, slice them in half horizontally to create thinner, even pieces. Season both sides generously with salt and pepper.
Heat the olive oil and butter in a large skillet over medium-high heat.
Add the chicken and sear for about 4–5 minutes per side, or until deeply golden brown and nearly cooked through. Make sure the chicken breasts or pieces are not crowding the skillet, you may need to do this in two batches.
Transfer the chicken to a plate and set aside. It's not finished cooking, it will continue to cook in the sauce until done.
Reduce the heat to medium.
Add the mushrooms and shallots to the same skillet. Cook for 5–7 minutes, stirring occasionally, until the mushrooms are golden brown and the shallots are softened. If the pan seems dry, add a small splash of olive oil or another teaspoon of butter.
Add the thyme and sage. Cook for about 30 seconds, just until fragrant. Last, add the garlic and cook for an additional 30 seconds.
Pour in the Marsala and use a wooden spoon to scrape up all the browned bits from the bottom of the skillet. Let the Marsala simmer for 1–2 minutes, allowing it to reduce slightly.
Pour in the chicken broth and bring to a gentle simmer.
Reduce the heat to medium-low and stir in the heavy cream. Simmer gently for 4–5 minutes, stirring occasionally, until the sauce begins to thicken.
Stir in the Parmesan cheese. Return the chicken and any juices from the plate back to the skillet. Spoon the sauce and mushrooms over the chicken.
Simmer gently for another 5-6 minutes, or until the chicken is cooked through and reaches 165 F in the thickest part. Taste and adjust with additional salt and pepper as needed.
Garnish with parsley and serve with mashed potatoes, egg noodles, polenta, or plenty of crusty bread for soaking up the sauce.