Halve each Andouille sausage lengthwise, then slice into thin half-moon pieces.
Heat a large nonstick skillet over medium heat. Add the sliced sausage and cook, stirring occasionally, until nicely browned, about 6–8 minutes.
Remove the cooked sausage with a slotted spoon to a bowl. Wipe the skillet clean and return it to the stovetop for grilling the sandwiches.
In a medium bowl, stir together the shredded extra-sharp cheddar (6 ounces) and 3 tablespoons softened butter until evenly combined.
Evenly spread the remaining 2 tablespoons butter on one side of each of the 8 bread slices.
On four bread slices, spread the cheddar–butter mixture onto the side that is not buttered.
Divide the cooked Andouille sausage evenly over the cheddar-topped slices.
Sprinkle the shredded Monterey Jack (6 ounces) evenly over the sausage on those four slices.
Spread the 4 teaspoons Dijon mustard on the non-buttered side of the remaining four bread slices. Place each mustard-coated slice Dijon-side down on top of the assembled slices to make 4 sandwiches (buttered side out).
Heat the wiped skillet over medium. Cook 2 sandwiches at a time: place them in the skillet, cook until the bottom is nicely browned, press gently with a spatula, flip, and cook until the second side is nicely browned. If the cheese is not fully melted, reduce heat to low and continue cooking until the cheese melts.
Repeat step 10 with the remaining 2 sandwiches. Remove sandwiches from the skillet, let sit briefly if desired, then serve.