Preheat the oven to 400˚F. Butter a 9x13 baking dish on the bottom and sides.

Slice the onions ½ inch thick, then separate each slice into individual rings. Keep the rings whole as much as you can, but a few broken ones are fine.

Whisk the beef base into the melted butter until smooth. To a very large bowl, add the onion rings and the butter mixture and toss until every ring is coated. Sprinkle the thyme, parsley, oregano, dry mustard, smoked paprika, garlic powder, salt, pepper and cayenne over the top. Set a second bowl of about the same size upside down over the first and shake until the onions are evenly coated.

Arrange the rings in the buttered dish so they lie mostly flat and intact. They don't need to be in perfect rows, but don't pile them in a heap.

Roast, uncovered, for 35 to 40 minutes, until the onions are very soft and starting to color at the edges. If they begin to brown too quickly, lay a sheet of foil loosely over the top for the remaining time.

Remove the dish from the oven. If there is liquid pooled in the bottom, sop it up with paper towels.

Scatter the mozzarella and cheddar over the onions, then the Parmesan. Bake for 15 to 20 minutes, until the cheese is melted and just starting to brown.

Switch the oven to broil and broil until the top is deep golden brown and bubbling, watching closely the whole time. Let rest for 5 minutes before serving.