Combine the flour, salt, and nutmeg in a large bowl and stir well. Whisk together the eggs and 2/3 cup water and add to the flour mixture, beating until just smooth. The texture should be the consistency of thick pancake batter. If too thick, whisk in 2 to 3 tablespoons more water.
Drop the mixture through a spaetzle maker or colander into a large pot of salted boiling water. Simmer until tender, 3 to 4 minutes. Drain and rinse under cold running water. (Spaetzle may be made a day in advance. Keep covered and chilled.)
Heat the butter and oil in a large skillet over high heat until hot. Add the spaetzle and cook, stirring often, until lightly browned and heated through, about 5 minutes. Add the chives and season with salt and pepper.