Heat the oven to 350°F with its rack in the middle position, and coat a 13-by-9-inch baking dish lightly with cooking spray. The coating should cover the bottom and sides without pooling.

Place each peeled onion on its side, slice crosswise into 1/4-inch rounds, separate the rings, and nest them from largest to smallest in the dish to form four stacks. The nested rings should resemble the sliced rounds, with smaller rings sitting inside larger ones rather than in a loose heap.

Combine the dried parsley, dry mustard, garlic powder, seasoned salt, oregano, kosher salt, cayenne, and black pepper in a small bowl, then blend the mozzarella, cheddar, and Parmesan in a medium bowl. Both mixtures should look evenly distributed so the onions receive consistent seasoning and cheese coverage.

Coat the onions with the seasoning mixture using a gentle toss, distribute the butter cubes over them, pour the broth into the bottom of the dish, and seal the dish tightly with foil. The butter should sit across the onions while the broth collects underneath for the covered bake.

Bake for 75 to 90 minutes, then lift away the foil carefully, test the onion centers, and spread the cheese mixture over the onions once they are tender. A fork or metal cake tester should slide easily through the center of a stack without meeting a firm, crunchy core.

Return the uncovered dish to the oven for 5 to 7 minutes to melt the topping. The shreds should merge into a melted layer with bubbling around the edges.

Switch on the broiler, watch the dish continuously for 1 to 2 minutes, and remove it when browned patches appear before finishing with fresh parsley. A few golden spots are enough; the entire surface doesn’t need to turn brown.