In a medium [large] sauce pot, combine the shiitake mushrooms and 4 cups [6 cups] water. Bring to a boil, cover, and cook, stirring occasionally, until softened, 3 to 5 minutes. Remove the mushrooms, reserving the soaking liquid in the pot. Discard the stems and coarsely chop the mushroom caps.
Return the water to a boil and season lightly with salt. Add the bucatini and cook until just tender, 10 to 12 minutes. Drain, reserving ½ cup [1 cup] pasta cooking water.
While the mushrooms soak and the bucatini cooks, start preparing the rest of the meal.

• Coarsely chop enough onion to measure ½ cup [1 cup].
• Trim the root ends from the scallions; thinly slice the scallions on the diagonal, keeping the white and green parts separate. Set aside the green parts for garnish.
• Cut a small corner from the ground pork packaging and drain off any excess liquid. Transfer to a plate; pat dry with a paper towel.

In a large frying pan over medium-high heat, warm 1 teaspoon [2 tsp] oil until hot but not smoking. Stir in the mushrooms, onion, and white parts of the scallions and cook until the onion and scallions start to soften, 1 to 2 minutes. Add the pork, season with salt and pepper, and cook, stirring to break up the meat, until lightly browned but not yet cooked through, 3 to 4 minutes.
Add the fermented black soybean paste and reserved pasta cooking water and bring to a boil. Reduce to a simmer and cook, stirring occasionally, until the sauce is thickened and the pork is cooked through, 3 to 5 minutes. Remove from the heat and season to taste with salt and pepper.
While the sauce cooks, prepare the remaining garnishes.

• Peel the cucumber, if desired; cut the cucumber in half lengthwise and scrape out the seeds with a spoon. Cut the cucumber into ¼-inch-thick matchsticks.
• Scrub or peel the daikon; thinly slice the daikon into rounds, then thinly slice the rounds into matchsticks.

Transfer the bucatini to individual bowls and top with the pork, vegetables, and sauce. Garnish with as much cucumber, daikon, and green parts of the scallions as you like and serve.