Heat the oven to 220°C / 425°F with a 25 cm / 10-inch cast-iron skillet inside, so the skillet heats for a full 10 minutes before the batter goes in.
Combine the eggs, cottage cheese, milk, flour, and salt in a blender. Blend on high for 45 to 60 seconds until completely smooth, no visible curds.
Dice the ham into small, even pieces if it isn't pre-diced, and shred the Gruyere if using a block.
Carefully remove the hot skillet from the oven using oven mitts. Add the butter and swirl it around the pan until melted and foaming, about 20 seconds.
Pour the batter into the skillet right away. Scatter the diced ham and half the Gruyere evenly over the top.
Return the skillet to the oven and bake for 20 to 22 minutes without opening the door, until the Dutch baby is puffed and deep golden at the edges and the center is just set.
Pull the skillet from the oven and immediately top with the remaining Gruyere and black pepper. Let it rest 2 to 3 minutes, it will start to deflate and that's normal.
Scatter chives over the top, slice into wedges, and serve right away while it's still puffed.