Bring a pot of water to a boil over high heat. Cook the noodles according to the instructions on the package. Drain well and rinse with cold water or submerge in an ice bath to stop the cooking. Drain immediately and set aside. You can perform this step closer to the end, as needed, depending on how quickly you want to serve the noodles.
Combine all the ingredients for the veggie broth in a large pot. Bring to a boil over medium-high heat. Reduce to medium-low heat to achieve a moderate boil. Keep at a moderate boil, stirring and adjusting the heat as needed, for the next 30-40 minutes. The vegetables should be extremely tender but not disintegrating.
Perform the same steps as described above for the ingredients in the Dashi Broth.
When the Veggie Broth has cooked enough as described above, turn off the heat for both of the broths. Strain out all the solid ingredients for each broth separately. Use a mesh strainer if needed. Place the boiled vegetables in a deep plate and allow to cool. Once cooled, strain out any remaining broth that you can, squeezing the vegetables as needed.
Combine both broths in a large measuring cup, in something close to a 1:1 ratio. If you have more than 700mL total, stop adding when you reach 700 mL. If you have less than 700 mL, add all of both broths into the measuring cup. Add ice so until the liquid level reaches 800mL.
Add the vinegar and maple syrup to the broth. Stir well. Add the salt, stir well, and taste. Add more salt as needed so that the broth is lightly salty to your taste.
Combine the Pepper Paste ingredients in a small bowl. Stir until evenly combined.
You can chill or partially freeze the broths at this point if desired.
Divide the noodles evenly among your serving bowls. Likewise with the broth and garnishes. Allow each person to add pepper paste to their noodles as desired. Enjoy immediately.