Heat 2 tablespoons oil in a large cast-iron skillet over high just until smoking. Add one-third of the mushrooms in a single layer, and cook, without stirring, until golden brown, about 3 minutes. Stir in 3 thyme sprigs, 2 crushed garlic cloves, ¼ teaspoon salt, and ¼ teaspoon pepper. Reduce heat to medium-high, and cook, stirring occasionally, until mushrooms are tender and browned all over, about 3 minutes. Add 1 tablespoon butter, 1 tablespoon parsley, and 1 teaspoon vinegar. Cook, stirring occasionally, until mushrooms are coated in butter, about 1 minute. Transfer mushroom mixture to paper towels to drain; discard cooked garlic and thyme.

Wipe skillet clean, and repeat process 2 times with remaining oil, remaining mushrooms, remaining thyme, remaining garlic, ½ teaspoon salt, ½ teaspoon pepper, 2 tablespoons butter, remaining parsley, and remaining vinegar. Place all mushrooms in a large bowl,and season with remaining ½ teaspoon salt and remaining ¼ teaspoon pepper. Keep warm or let cool to room temperature.

Preheat oven to 350°F. Place brioche slices in a single layer on a wire rack set inside a baking sheet. Bake in preheated oven until lightly toasted, about 8 minutes per side. Brush with remaining 2 tablespoons softened butter.

Spread cheese evenly over brioche toasts. Spoon mushroom mixture evenly over toasts; garnish with parsley, pickled onion slices, frisée, and chives. Drizzle with lemon juice and extra-virgin olive oil.