Make Honey Dijon: In a small ramekin, whisk together the whole-grain Dijon mustard and pure honey until well incorporated.
Prep Bread: Place sourdough slices on a board. Spread the honey Dijon sauce evenly across the interior sides of all four slices. Generously spread softened butter on the outside faces of the bread.
Assemble Sandwiches: On the bottom two slices, arrange half of the sliced sharp white cheddar. Layer folds of smoked turkey breast over the cheese, then arrange thin Honeycrisp apple slices and fresh baby arugula on top. Finish with the remaining cheddar cheese and cap with the top bread slices (butter-side up).
Grill & Press: Preheat a panini press or heat a cast-iron grill pan over medium heat. Place sandwiches in the pan (if using a skillet, press down firmly with a spatula or weigh down with a heavy pan). Grill for 3 to 4 minutes per side until the bread is deeply toasted with crisp ridges and the cheese is fully melted and bubbling.
Slice & Enjoy: Transfer to a cutting board, let rest for 60 seconds, slice diagonally in half, and serve hot with soup or chips.