Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.
Spread Dijon mustard on one side of each slice of gluten-free bread. Layer the ham and half of the Gruyère cheese on one slice, then top with the other slice, mustard side down.
In a skillet over medium heat, melt 1 tablespoon of butter. Add the sandwich and cook until golden brown on one side, about 3-4 minutes. Carefully flip and cook the other side until golden brown and the cheese begins to melt, about another 3-4 minutes.
Transfer the sandwich to the prepared baking sheet. In a small bowl, whisk together the egg, milk, salt, and pepper. Pour the egg mixture over the sandwich, allowing it to soak in.
Sprinkle the remaining Gruyère cheese on top of the sandwich. Bake in the preheated oven for 10-12 minutes, or until the egg is set and the cheese is bubbly and golden.
Remove from the oven and let cool slightly. Serve warm, garnished with chopped parsley if desired.