Combine 2 tablespoons each butter and olive oil in a wide skillet. Place over medium high heat.
When butter is melted, add onions so they lie flat in the pan. Let them cook undisturbed until they are slightly charred, about 7 minutes.
Reduce heat, flip onions over, and cook until they are softened, another 10 minutes. Transfer to a warm plate.
Wipe out skillet. Combine remaining butter and oil. Place back over medium heat.
When butter is melted, add garlic and cook for 30 seconds. Add mushrooms, stir and cook until mushrooms soften, 4–6 minutes. Season with salt and stir in thyme leaves. Keep warm until polenta is cooked.