In a skillet, combine 1 ½ tablespoons of butter with the olive oil over medium-low heat. Once the butter has melted, toss in the diced onions, a pinch of salt, pepper, minced thyme, and one teaspoon of fresh rosemary. Sauté, stirring frequently, until the onions soften and start to brown, roughly 10 minutes. This step is crucial, as the caramelization brings out the onions' natural sweetness.

Next, stir the brown sugar into the onions and cook for an additional minute. This step amplifies that sweet flavor, balancing the savory notes from the herbs. Once done, transfer the onions to a plate and set them aside.

Spread a thin layer of mayonnaise on one side of two slices of bread. The mayo will create a rich, golden crust when grilled.

In the same skillet, add the remaining butter and the last teaspoon of rosemary. Heat over medium heat, swirling to coat the bottom of the skillet with the melted butter. This infuses more flavor into your grilled cheese.

Place the two slices of bread into the skillet, mayonnaise side down. Grill for about 2-3 minutes, until golden brown and crunchy. This is where the magic starts, as the bread gets crispy and delicious.

As soon as the bread hits the skillet, pile a mix of both Gruyere and white cheddar cheese on top of each slice. Once the cheese begins to melt, add a generous sprinkling of caramelized onions over one of the slices.

Place the other slice of bread, cheese side down, onto the onion-covered slice. Carefully press down to encourage the melting. Cook on low heat for about 5 minutes, flipping once, until the cheese is fully melted, and the bread is beautifully toasted.

Remove from the skillet, slice in half, and enjoy! The combination of gooey cheese with the sweet, caramelized onions is a match made in culinary heaven.