Cut the zucchini into half-moons and spread on a clean kitchen towel. Lightly salt and let sit for 10 minutes to draw out moisture, then pat very dry. Quarter the mushrooms and dice the onion.

Heat 1 tablespoon olive oil in a 12-inch skillet over medium-high heat until shimmering. Add mushrooms in a single layer and cook undisturbed for 4-5 minutes until deeply browned on one side. Stir and cook 2 minutes more, then transfer to a plate.

Add another tablespoon of oil to the pan. Add zucchini in a single layer and cook without moving for 3-4 minutes until golden brown underneath. Flip and cook 2-3 minutes more until tender but not mushy. Transfer to the plate with mushrooms.

Reduce heat to medium. Add remaining tablespoon of oil and the butter. Once foaming, add onion and cook 4 minutes until softened and lightly golden. Add garlic, thyme, and red pepper flakes; cook 1 minute until fragrant.

Return mushrooms and zucchini to the pan. Toss everything together and cook 2 minutes to meld flavors. Remove from heat, stir in lemon juice and Parmesan. Taste and adjust salt and pepper. Sprinkle with parsley and serve hot with extra cheese on the side.