Melt 1 tablespoon unsalted butter with 1 tablespoon olive oil in a heavy-bottomed pot over medium-high heat. Slice 2 yellow onions 1/4-inch thick from root to tip and add to pot. Season with salt and cracked black pepper, and spread in an even layer. Cover and turn heat to medium-low. Sweat, steam and soften the onions, 10 to 15 minutes. Uncover and continue cooking until caramelized, 25 to 30 minutes, stirring frequently. Add 1 tablespoon dry sherry and scrape up any bits from the bottom of the pan. Stir in 1/2 teaspoon finely chopped fresh thyme and season with salt and pepper. Preheat the griddle to medium-low heat. Put 1 slice white American cheese on 4 pieces country white bread and 1 slice yellow American cheese on 4 more bread pieces. Add some onions. Close and butter both sides with softened salted butter. Place on the griddle; cover with a metal bowl and let the sandwiches get a nice golden brown, 2 to 3 minutes. Flip, cover again and cook until the cheese is melted and the sandwiches are golden brown, 2 to 3 minutes.
Food Networks Kitchen Tip: Be patient! Let the onions get nice and brown to coax out maximum flavor.
From Food Network Kitchens; after further testing and to ensure the best results this recipe has been altered from what was in the actual episode.