Prep the vegetables: Clean the mushrooms with a damp towel and slice them. Finely chop the onion and mince the garlic.
Heat a large skillet over medium-high heat. Add the olive oil and 2 tbsp of the butter. Add the sliced mushrooms in an even layer and cook, stirring occasionally, until they release their liquid and turn deep brown, 8–10 minutes.
Reduce the heat to medium. Push the mushrooms to the sides, add the remaining 2 tbsp butter to the center, then add the onion. Cook until the onion is soft and translucent, 5–6 minutes. Stir in the garlic and dried thyme and cook until fragrant, about 30 seconds.
Sprinkle the flour over the mushrooms and onions. Stir constantly to coat and cook the flour until it smells nutty, 2–3 minutes.
Pour in the dry white wine and scrape up any browned bits. Simmer until mostly evaporated, 1–2 minutes.
Gradually whisk in the vegetable broth until smooth, then stir in the soy sauce.
Bring to a gentle simmer and cook, stirring occasionally, until the gravy is glossy and thick enough to coat the back of a spoon, 8–12 minutes.
Stir in the kosher salt and black pepper, then taste and adjust seasoning as needed. Serve hot.