Heat a large skillet over medium-high heat; swirl in half of oil and half of butter. Add half of mushrooms and half of thyme; cook, stirring occasionally, until mushrooms have released most of their moisture and become golden, about 14 minutes. Add half of cognac and cook, scraping up browned bits from bottom of pan. Season with salt and pepper. Transfer to a bowl.
Repeat with second half of all ingredients. Return first batch to skillet until heated through before serving.