Melt the butter with the olive oil in a heavy-bottomed pot over medium-high heat. Add the onions, season with salt and pepper and spread the onions in an even layer. Cover the pot and turn down the 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 the sherry and scrape up any bits from the bottom of the pan. Stir in the thyme and season with salt and pepper.
Preheat a griddle to medium-low heat. Put 1 slice white American cheese on 4 slices country white bread and 1 slice yellow American cheese on 4 more bread slices. Divide the onions between half the bread, then close and butter both sides. Place the sandwiches on the griddle; cover with a metal bowl (careful, it will get hot) and let the sandwiches get nice and 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.
Photograph by Levi Brown
Courtesy of Food Network Magazine