Combine all the flour and 3/4 teaspoon salt in a large bowl. In a separate bowl, whisk together the eggs and milk, then add to the flour, whisking until smooth. Add 1/4 cup of mustard and oil, and mix well. Prepare an ice bath in a large bowl and set aside.

Bring 4 quarts of water seasoned with the remaining 1 teaspoon of salt to a rolling boil in a large Dutch oven or large, heavy pot. Pour half of the batter into a spaetzle maker or a medium colander set over the boiling water. With a rubber spatula, force the batter through the holes of the spaetzle maker or colander into the water. (The water will rise and foam quickly.) Cook the dumplings, stirring, until they rise to the surface, about 5 minutes. Transfer with a long-handled spoon or strainer to the ice bath. Repeat with the remaining batter. Drain and place in a large bowl. (The spaetzle can be prepared to this point up to 2 hours in advance, and kept tightly covered in the refrigerator until ready to assemble.)

Melt the butter in a large non-tick skillet over medium-high heat. When the butter is foamy, add the remaining tablespoon of mustard and swirl to blend. Add the spaetzle, parsley, and lemon juice, and stir to mix. Cook until the spaetzle is warmed through and the flavors are well blended, about 2 minutes.

Remove from the heat and serve immediately. This is a great side dish to accompany either pork, sausage, or chicken.