Heat a large saute pan or cast iron pan over medium heat. Add the olive oil and the mushrooms. Season with salt and pepper, to taste, and add the thyme and the garlic and saute. Cook until the mushrooms are tender and a lot of the liquid has evaporated from the bottom of the pan, 6 to 8 minutes.
Add the Marsala and cook until the flavor of the alcohol has mellowed considerably, 3 to 5 minutes. Remove and discard the sprigs of thyme and the garlic. Shut the heat off and stir in the spinach leaves so they wilt slightly. Season with salt and pepper, to taste. Transfer to a serving bowl and serve.