Heat a large skillet over medium low. To make each sandwich, spread one side of a bread slice with butter, ensuring that you cover all the way to the edges. Spread the other side of the buttered slice with 1 teaspoon Dijon mustard, then layer on half of the cheese, 2 strips of cooked bacon, tomato slices, and a small handful of arugula. Spread another slice of bread with Dijon mustard, then place it mustard-side down on top of the arugula. Butter the remaining topside of the bread. Repeat for the second sandwich.
Once the skillet is warm, cook the sandwiches on each side, until the bread is golden and the cheese melted, about 3 to 5 minutes per side. Be patient and resist the urge to turn up the skillet heat—low and slow will ensure a golden toast and gooey center. Transfer to a plate, let cool a few minutes, slice, and enjoy.