In a large skillet over medium-high heat, melt the butter and olive oil together. When the butter melts and foams, add the sliced onions. Season with salt, pepper, and thyme. Spread the onions into an even layer and cover the pan.

Reduce heat to medium-low and let the onions steam for 10 to 15 minutes until softened. Remove the lid and continue cooking for another 25 minutes, stirring frequently, until the onions are jammy and deep golden brown. Do not rush this. Low and slow is the only way.

Once the onions are caramelized, add the white wine, beer, or beef stock to deglaze — scrape up all the browned bits from the bottom of the skillet. Those bits are flavor. Remove from heat and set aside.

Preheat a griddle or large skillet to medium-low heat. Butter both sides of each bread slice. Place one slice butter-side down and pile it with Gruyere cheese, Asiago, and a generous heap of caramelized onions. Top with the second slice of bread, butter side up.

Cover the sandwiches with a lid or an inverted bowl. Cook for 2 to 3 minutes until the bottom is golden brown, then flip carefully. Cover again and cook another 2 to 3 minutes until the cheese melts completely and both buttered sides are deep golden brown.

Move to a cutting board, cut on the diagonal, and serve immediately.