Cook the bacon in a skillet over medium heat until crisp, then drain on paper towels.
Preheat a panini press or warm a large skillet over medium heat.
Stir the mayonnaise and Dijon together in a small bowl.
Slice each bread portion horizontally. Spread the mayo mixture on the cut sides.
Layer chicken, cheese, bacon, and tomato on the bottom halves. Season with salt and pepper, then add the greens.
Close the sandwiches and butter the outsides of the bread.
Press in the panini maker until the bread is golden and cheese melts, 3–5 minutes. If using a skillet, cook 3–4 minutes per side, pressing with a heavy pan.
Rest 1 minute, then slice and serve warm.