Cut the stale Kaisersemmel or rolls into cubes no larger than 1/2 inch (1.25 centimeters). Place in a large bowl.
Pour the warm milk over the bread cubes and set aside.
In a large skillet, melt the 2 tablespoons (28 grams) butter over medium heat.
Once melted, add the finely chopped onion and cook, stirring often, until softened and translucent.
Add the garlic and cook just until fragrant, 30 seconds to 1 minute.
Remove from heat, then stir in the salt, nutmeg, pepper, and parsley.
Add the mixture to the bowl with the bread cubes.
Stir briefly to mix, then add the eggs, shredded cheese, and flour.
Using your hands, combine the ingredients and briefly knead to bring everything together. Do not overmix. Set aside at room temperature for 20 minutes to allow the flavors to settle.
Lightly moisten your hands with water and divide the mixture into 9 equal pieces (or 6 for larger dumplings or up to 12 for smaller pieces) and form each into a patty about 1 inch (2.5 centimeters) thick.
In a large skillet, melt the clarified butter over medium low heat.
Add the patties in batches, taking care to not overcrowd, and cook until golden brown.
Carefully flip and cook the other side until golden brown and heated through.
Remove to a plate and repeat with remaining Kaspressknödel.
Serve immediately either in a beef broth with a sprinkling of chives or alongside a salad with a drizzle of brown butter and a sprinkling of chives.