Arrange a rack in top third of oven and heat oven to 400°F.

Melt butter in a large skillet over medium heat. Add onions, thyme, 1⁄2 teaspoon salt, and several grinds of pepper. Cook, stirring frequently, until onions have reduced by more than half and are very soft and deep golden brown, about 28 to 30 minutes. Reduce heat if onions start to burn.

Meanwhile, remove and discard stems from mushrooms. Use a spoon to gently scrape out and discard gills to make room for filling. Brush mushroom caps on both sides with 2 tablespoonsolive oil and season with 1⁄2 teaspoon salt and several grinds of pepper. Place mushrooms on a rimmed baking sheet, stem side down, and roast until they just begin to soften, about 10 minutes.

Once onions are caramelized, pour in the wine. Scrape up any browned bits on the bottom of the pan and continue to stir until wine has evaporated, about 1 minute. Remove pan from heat and stir in 1 teaspoon vinegar.

Once mushrooms are just softened, remove from oven and flip mushrooms over with a flat spatula. Divide caramelized onions evenly among mushroom caps, then sprinkle with Gruyère.

Roast until mushrooms are tender and cheese just starts to melt, 5 to 7 minutes, then broil on high until cheese browns, 2 to 3 minutes.

Meanwhile, heat 1 tablespoon olive oil in a small skillet over mediumheat until shimmering. Add breadcrumbs, garlic, a pinch of salt, and a few grinds of pepper. Toast, stirring frequently, until golden brown, 3 to 5 minutes. Transfer to a small bowl.

Whisk together the remaining 2 tablespoons olive oil, 2 tablespoons vinegar, Dijon, a generous pinch of salt, and several grinds of pepper in a large bowl. Add greens and toss to coat. Taste and season with additional salt and pepper as needed. Divide among individual shallow bowls or plates.

Top each salad with a stuffed mushroom and sprinkle with toasted breadcrumbs.