Preheat an oven to 400 F.
Preheat the skillet. Drop 2 tablespoons of butter into your skillet and transfer it to the oven for 5 minutes to preheat. Keep an eye on the skillet so that the butter doesn't burn.
Make the batter. Combine the milk, eggs, flour, dijon, salt, and nutmeg in a blender. Blend until smooth, then add the chives, and blend for another 10 seconds.
Bake. Pour the batter into the preheated skillet and transfer to the oven for 20 minutes, until golden brown and puffed.
Make the bechamel. While the dutch baby bakes, melt 1 tablespoon of butter in a small saucepan. Add 1 tablespoon of flour and cook while whisking constantly for 1 minute. Slowly stream in the milk while whisking and bring to a gentle simmer. Once thickened, stir in the whole grain mustard, nutmeg, and season with salt and pepper. Set aside.
Melt the cheese. After 20 minutes, remove the dutch baby from the oven and top with the gruyere. Return to the oven for 5 minutes, until the cheese is melted.
Fry the ham and eggs. In a nonstick skillet, heat 1 tablespoon of olive oil over medium heat. Once the pan is hot, add the ham slices and cook for 1-2 minutes, until browned. Remove the ham from the pan and fry the eggs to your desired doneness.
Assemble. Remove the dutch baby from the oven and let it cool for 5 minutes. Top with the bechamel, ham, and fried eggs before serving.