Combine Dressing ingredients in large bowl. Whisk until well blended; refrigerate until ready to use.
Attach Pasta Press Accessory fitted with Bucatini plate to KitchenAidⓇ Stand Mixer. Shape each piece of dough into 1-inch diameter log. Turn mixer to Speed 8. Working with one log of dough at a time, feed dough into hopper and use the swing-arm wire dough cutter to cut pasta into 8-inch lengths as it exits the press. Carefully arrange pasta on rimmed baking sheet lightly dusted with flour.
Bring 4 quarts water to a boil in large pot; season generously with salt. Add pasta and cook 3 minutes or until al dente. Drain pasta; rinse with cold water.
Meanwhile, combine remaining ingredients with Dressing. Add cooled pasta; gently toss until combined.