In a deep skillet over medium-high, melt 1 tablespoon of the butter in the oil. Add onion, cover and cook, stirring once or twice, until softened, about 5 minutes. Uncover and cook over medium, stirring occasionally, until onion is very tender and caramelized, about 35 minutes longer. Add water to the skillet as necessary, ¼ cup at a time, to prevent onion from scorching. Season caramelized onions with salt and pepper.
Spread 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 onion, Gruyère and pickles and close sandwiches.
Preheat a skillet or panini press. Grill sandwiches over low heat until bread is toasted and cheese is melted, about 10 minutes. If using a skillet, press sandwiches with a spatula and flip them halfway through. Cut in half and serve right away.
