Preheat the oven to 350 degrees.
In a medium bowl, whisk together the eggs and milk.
Heat a non stick skillet over medium low heat. Melt the butter then add the egg mixture. Drag a spatula through the eggs as they cook, keeping them moving pretty continuously until curds start to form. When the eggs are in soft curds but still look a bit wet, remove from heat. Sprinkle with cheese and stir. Season to taste with salt and pepper.
Meanwhile, whisk together all of the sauce ingredients and set aside.
To assemble the sandwiches, layer the bottom half of the croissants with ham, cheese, and egg mixture. Slather the top half with the sauce and sandwich together.
Place on a baking sheet and bake for 8-10 minutes, or until cheese is melted and gooey. Serve warm.