In a large bowl, mix together flour, 4 eggs, about 1/4 teaspoon salt, and milk with a wooden spoon until thoroughly combined.
Bring a large pot of water to a boil. Pour in 1 teaspoon of oil.
Transfer the batter to a spaetzle maker or a colander with large holes and place over the boiling water. Slide the press back and forth or use a wooden spoon to press the batter through the holes of the colander. Stir gently to remove any pieces from the bottom of the bot. Cook until they float to the top, remove with a strainer, rinse with cold water, and drain.
In a medium bowl, beat together remaining 4 eggs. Season with salt.
In a large pan, drizzle 3 tablespoons oil over medium heat. Add the cooked dumplings. Once heated through, pour in eggs. Stir continuously until all the eggs are cooked.