I'm born in the SW part of Germany, where Spätzle come from.
a) I'd never, ever use milk, so steep the thyme in water. The basic recipe uses only flour, salt, egg and a bit of cold water.
b) You can push the batter trough a potato ricer, this makes longish noodles ("Spätzle") and is a lot quicker. The method described here yields "Knöpfle" ("buttons")
c) Always beat the batter a bit, it should become somewhat "stretchy" or elastic. A wooden spoon is best.