In a deep skillet, melt 1 tablespoon of the butter in the oil. Add the onion, cover and cook over high heat, stirring once or twice, until softened, about 5 minutes. Uncover and cook over moderate heat, stirring occasionally, until the onion is very tender and caramelized, about 25 minutes longer. Add water to the skillet as necessary, 1/4 cup at a time, to prevent the onion from scorching. Season the caramelized onion with salt and pepper.

Spread the bread with the remaining 2 tablespoons of butter and arrange, buttered side down, on a work surface. Spread a thin layer of mustard on each slice. Top half of the slices with the onion, Gruyere, and pickles and close the sandwiches.

Preheat a skillet or panini press. Grill the sandwiches over low heat until the bread is toasted and the cheese is melted, 10 minutes. If using a skillet, press the sandwiches with a spatula and flip them halfway through. Cut in half and serve right away.