The best grilled cheese is the gooiest grilled cheese, says author Laura Werlin: "After you bite into it, the cheese should stretch out past your face as far as your arm will reach. Otherwise, it's just not right."
In a large nonstick skillet, cook the sliced sausages over moderate heat, turning once, until browned, about 8 minutes. Transfer the sausages to a plate and wipe out the pan.
In a bowl, stir the butter with the shredded cheddar cheese just until combined. Spread the cheddar-butter on one side of each slice of bread; arrange them buttered side down on a work surface. Spread the mustard on the bread and sprinkle the Monterey Jack over the mustard. Top 4 slices with the sausages, halved cornichons and the sliced cheddar. Close the sandwiches, pressing lightly and tucking in any stray cheese.
Place the sandwiches in the skillet, cover and cook over moderately low heat until golden on the bottom, 3 minutes. Press with a spatula to flatten slightly, then flip the sandwiches, cover and cook until browned on the bottom, 3 minutes longer; lower the heat if the bread is browned before the cheese melts completely. Transfer the sandwiches to a cutting board and let rest for 2 minutes, then cut them in half and serve with cornichons.