Preheat oven to 325 degrees F. Heat oil in a large nonstick skillet over medium-high heat. Add mushrooms in an even layer; cook, undisturbed, until browned on the bottom, about 4 minutes. Stir and continue to cook, stirring occasionally, until browned all over, about 5 minutes. Add onion; cook, stirring occasionally, until beginning to soften, about 4 minutes. Stir in garlic and thyme; cook, stirring, until fragrant, about 2 minutes. Add spinach; cook, stirring constantly, until just wilted, about 2 minutes. Remove from heat.