Melt 2 tablespoons of butter with 1 tablespoon of olive oil in a wide heavy-bottomed skillet or Dutch oven over medium heat. Add the sliced onions, sea salt, and black pepper. Toss to coat evenly.

Lower heat to medium-low. Cook onions gently for 25 to 30 minutes, stirring every few minutes, until they collapse into deeply golden, soft, jammy caramelized strands. If the onions begin to scorch, reduce heat to low and add a tablespoon of water.

Add the minced garlic and fresh thyme leaves; cook for 1 minute until fragrant. Pour in the beef bone broth and Worcestershire sauce, using a wooden spoon to deglaze and scrape up all the rich browned bits on the bottom. Simmer for 2 minutes until the liquid is fully absorbed into the jammy onions. Remove from heat.

In a small bowl, mix the 3 tablespoons softened butter and 1 tablespoon mayonnaise until smooth. Spread a thin, even layer of this butter-mayo mixture onto one side of each sourdough bread slice (this will be the crispy outside).

Lay 4 bread slices butter-side down on a clean cutting board. Layer each with a quarter of the shredded Gruyère cheese, a generous mound of warm caramelized onions, and a layer of shredded sharp white cheddar (cheese on both bottom and top locks the bread together as it melts). Top with the remaining bread slices, butter-side facing up.

Heat a large cast iron skillet or griddle over medium-low heat. Place the sandwiches into the skillet. Cook for 3 to 4 minutes per side, pressing down gently with a flat spatula, until the sourdough is deeply golden-brown and shatteringly crisp, and the cheeses are completely melted and bubbling at the seams.

Transfer to a cutting board and let rest for 1 full minute so the molten cheese sets slightly. Slice diagonally in half to reveal the epic cheese pull, and serve immediately alongside warm French onion au jus dipping broth or crisp sea salt chips!