Bring milk and 4 thyme sprigs to a boil in a small saucepan. Remove from heat. Discard thyme. Transfer milk to a blender; add parsley, thyme leaves, and eggs. Blend until combined.
Whisk flour and 1 1/2 teaspoons salt in a medium bowl. Gradually whisk in milk mixture until smooth.
Bring a medium saucepan of salted water to a boil. Working with about 1/4 cup batter at a time, press batter through 1/4-inch holes of a colander or spaetzle maker into boiling water. Cook until spaetzle float to top, 1 to 2 minutes. Using a slotted spoon, transfer to an ice-water bath. Drain.
Melt butter in a medium nonstick skillet over medium-high heat. Add half of the spaetzle; cook, stirring occasionally, until golden, about 6 minutes. Transfer to a bowl. Repeat. Season with salt and pepper. Garnish with remaining thyme.