In a small bowl, combine two tablespoons flour and ½ teaspoon salt. Add the sliced half of the onion and toss to coat.
Line a medium plate with paper towels and set it by the stove. In a medium skillet over medium-high heat, heat the oil until shimmering. Add the sliced onions (leave the excess flour behind) and fry, using tongs to stir occasionally, until crisp and golden-brown, 9–10 minutes. Transfer to the lined plate and transfer the oil to a heatproof bowl to cool completely before discarding. Wipe the skillet clean with paper towels and set aside.
Bring a large pot of well-salted water to a boil over high heat.
Meanwhile, return the skillet to the stove and set over medium heat. Add the butter. When it has melted and the foam begins to subside, add the chopped half of the onion and cook, stirring occasionally, until soft and translucent but not yet browned, 8–9 minutes. Remove from the heat and set aside.
In a large bowl, whisk the eggs until very frothy. Add the remaining 1¾ cups flour, 1 teaspoon salt, and the nutmeg, and whisk to a uniform, thick batter. Continue whisking until bubbles slowly rise to the surface when you stop whisking, about 2 minutes more.
When the water begins to boil, place a spätzle maker or a heat resistant colander with large holes over the pot. Pour the batter onto the spätzle maker or colander. Working quickly, use a bowl scraper or rubber spatula to force the all of the batter through the holes. Immediately stir the water gently to separate any stuck pieces. When the water returns to a boil, cook the spätzle for 2 minutes, then drain them and rinse well under cold water.
Return the large skillet with the chopped onions to the stove and set it over medium-high heat. When the onions begin to sizzle, add the spätzle and cheese. Stir until the spätzle is hot and the cheese is fully melted. Season with additional salt to taste, then transfer to a large bowl or 4 small plates. Topped with the crisped onions and chives, sprinkle with freshly ground black pepper, and serve immediately.