Heat olive oil in a large saucepan over medium heat.
Sauté onion until translucent, then add garlic and thyme for 1 minute.
Add mushrooms and cook until golden brown, releasing their juices.
Pour in vegetable broth, bring to a simmer for 10 minutes.
Blend soup with immersion blender or in batches in a regular blender until creamy.
Stir in plant milk, season with salt and pepper.
Simmer 5 more minutes, adjusting thickness if needed.
Serve hot, optionally garnished with thyme sprigs or a drizzle of olive oil.