Slice the 3 Andouille sausages: cut each sausage in half lengthwise, then slice crosswise into thin half‑moon pieces.
Heat a large nonstick skillet over medium. Add the sliced sausage and cook, stirring occasionally, until nicely browned, about 6–8 minutes. Use a slotted spoon to transfer the cooked sausage to a bowl. Wipe the skillet clean and leave it on the stove.
In a medium bowl, stir together the 6 ounces shredded extra‑sharp cheddar and 3 tablespoons of the softened butter until evenly combined.
Spread the remaining 2 tablespoons butter on one side of each of the 8 bread slices.
Take four bread slices and, on the non‑buttered side of each, spread an even layer of the cheddar‑and‑butter mixture.
Divide the cooked Andouille sausage evenly over the cheddar layer on those four slices. Sprinkle the 6 ounces shredded Monterey Jack evenly on top of the sausage.
On the non‑buttered side of the remaining four bread slices, spread the Dijon mustard. Place each mustarded slice on top of a filled slice with the Dijon side down so the buttered sides are facing outward.
Preheat the wiped skillet over medium. Add two sandwiches to the skillet (buttered side down). Cook until the bottom is nicely browned, then press gently with a spatula and flip to brown the other side. If the cheese is not fully melted after browning both sides, turn the heat to low and cook until the cheese melts.
Remove the finished sandwiches and repeat step 8 with the remaining two sandwiches. Serve hot.