Weigh or lightly spoon 4.5 ounces flour into a dry measuring cup, and level with a knife. Sift together flour and 1/4 teaspoon salt. Combine milk and eggs in a medium bowl; stir with a whisk. Add flour mixture to egg mixture, stirring until well combined. Let stand 10 minutes.
Bring 2 quarts water to a boil in a large saucepan. Hold a colander with large holes (about 1/4 inch in diameter) over boiling water; spoon about 1/2 cup dough into colander. Press dough through holes with a rubber spatula (droplets will form spaetzle); set colander aside. Cook 3 minutes or until done (spaetzle will rise to the surface). Remove with a slotted spoon; drain in a strainer (spaetzle will stick to a paper towel). Repeat procedure with remaining dough.
Combine porcini mushrooms and 1/2 cup boiling water in a small bowl; cover and let stand 30 minutes. Drain mushroom mixture in a colander over a bowl, reserving 1/4 cup liquid. Rinse and chop mushrooms.
Heat oil in a Dutch oven over medium-high heat. Add onion and garlic to pan; sauté 2 minutes. Add reserved porcini mushrooms, cremini mushrooms, and button mushrooms to pan; sauté 15 minutes or until moisture almost evaporates. Stir in 2 tablespoons flour and paprika. Add wine to pan; cook 1 minute or until liquid is absorbed. Stir in reserved mushroom liquid and broth; bring to a boil. Reduce heat, and simmer 15 minutes, stirring occasionally. Add 1/4 cup parsley, vinegar, 1/4 teaspoon salt, and pepper to pan; cook for 1 minute. Add spaetzle to pan; cook 2 minutes or until thoroughly heated. Garnish with remaining 1/4 cup parsley.