Step 1

In a large skillet over medium-high heat, heat the oil until shimmering. Add the onion and salt and cook, stirring often, until they start to soften and brown, 5 to 6 minutes. Reduce the heat to medium-low and cook, stirring occasionally, until the onions are deeply browned and sticky, 30 to 45 minutes. Stir in the vinegar. Transfer the onions to a small bowl.

Step 2

In a small bowl, mix 1 tablespoon butter with the miso paste and mustard. Spread one side of each bread slice with the remaining 2 tablespoons butter. Spread the other sides with the miso butter.

Step 3

Return the skillet to medium-low heat. Place two of the bread slices, plain-buttered-side down, in the skillet. Divide the cheese and onions evenly between them and top with the other bread slices, plain-buttered-side up. (You want the miso butter on the inside of the sandwiches.) Cook until the bread is golden brown, 5 to 7 minutes. (Resist the urge to press the sandwich with a spatula.) Flip the sandwiches over and cook on the other side until the bread is golden brown and the cheese is melted, 5 to 7 minutes. Transfer to plates and serve with cornichons, if desired.