1. Prep the Steaks: Take the steaks out of the refrigerator about 30 minutes before cooking to bring them to room temperature—season steaks on both sides generously with salt and black pepper. Preheat your oven to 400°F (200°C).
2. Sear the Steaks: Heat 2 tablespoons of olive oil in a heavy, oven-safe skillet over high heat. Once the oil is hot, add the steaks to the hot skillet. Cook for about 2-3 minutes on each side to get a nice sear. 
3. Transfer the skillet to the preheated oven and cook for an additional 5-7 minutes for medium-rare. Adjust cooking time based on your desired level of doneness- 55C/130F for medium rare and 57C/135F for medium. 
4. Remove the skillet from the oven and transfer the steaks to a plate. Tent them with aluminum foil to rest.

Make the Marsala Mushroom Sauce:

1. Add olive oil and butter in the same skillet you used for the steaks. Heat over medium-heat. Add the chopped shallot and cook until it becomes translucent.
2. Stir in the minced garlic and thyme, and cook for another minute.
3. Add the sliced mushrooms and cook for about 5 minutes, or until they are browned. Sprinkle the mushrooms with flour and stir to coat. 
4. Pour in the Marsala wine and deglaze the pan, scraping up any browned bits from the bottom of the skillet.
5. Add the beef or chicken broth and simmer for about 5 minutes until the sauce thickens. Reduce the heat to low and stir in the heavy cream. Simmer for  5-7 minutes until the sauce thickens further. Stir occasionally. 
6. Season the sauce with salt and black pepper to taste.

Serve:

1. Slice the rested steaks against the grain into thick slices.
2. Place steak on a plate and spoon the Marsala mushroom sauce. Garnish with additional fresh thyme or chopped parsley if desired. 

Enjoy your homemade Steak Marsala with Mushroom Sauce! This dish pairs wonderfully with polenta, mashed potatoes, or a side of green vegetables.