The secret to mushrooms that are beautifully browned and not watery is taking a hands-off approach. Stirring them as they caramelize releases the mushrooms' liquid and causes them to steam, so leaving them alone is the way to go.
Heat oil in a large skillet over medium-high heat. Cook mushrooms, cap sides down, without stirring, until caps are caramelized, 4 to 5 minutes.
Reduce heat to medium. Toss mushrooms, and season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook until mushrooms are well browned, about 5 minutes more. Add garlic, butter, and thyme. Cook until garlic is golden and fragrant, about 2 minutes. Remove skillet from heat. Add wine. Return skillet to heat, and cook until wine is evaporated, about 4 minutes. Garnish with parsley. Serve warm.