Position oven rack in upper third; preheat to 450°F.

Whisk 2 tablespoons oil, 1 tablespoon melted butter, 1 tablespoon thyme and ½ teaspoon each salt and pepper together in a small bowl. Arrange sliced onions in an even layer in a broiler-safe, 9-by-13-inch metal baking pan (don’t use glass or ceramic, which might shatter). Brush the butter mixture all over the onions. Do not wipe out the bowl; set aside. Roast the onions, flipping once halfway, until lightly browned on both sides, about 30 minutes.

Meanwhile, in the reserved bowl, combine ⅓ cup panko and the remaining 1 tablespoon melted butter and ½ teaspoon thyme; toss until evenly coated. Transfer to a small nonstick skillet; cook over medium-high heat, stirring occasionally, until fragrant and golden brown, 2 to 3 minutes. Remove from heat and allow to cool until ready to use.

Flip the onions once more. Carefully pour ½ cup broth and 1 tablespoon vinegar into the baking pan. Roast until the broth is slightly reduced and the onions are very soft, about 10 minutes more. Remove from oven.

Increase oven temperature to broil. Combine ½ cup each Gruyère and mozzarella in a small bowl; evenly sprinkle over the onions. Broil until the cheese is melted and bubbling, about 1 minute. Top with the reserved panko mixture. Garnish with thyme leaves, if desired. Serve with any sauce left in the pan.