Toss pork, 2 teaspoons water and baking soda in bowl until thoroughly combined. Let stand for 5 minutes. Whisk 1/2 cup water, miso paste, soy sauce, hoisin and molasses together in second bowl.
Pulse white and light green scallion parts, garlic and ginger in food processor until coarsely chopped, 5 to 10 pulses, scraping down sides of bowl as needed. Add mushrooms and pulse until mixture is finely chopped, 5 to 10 pulses.
Heat oil and pork mixture in large saucepan over medium heat for 1 minute, breaking up meat with wooden spoon. Add mushroom mixture and cook, stirring frequently, until mixture is dry and just begins to stick to saucepan, 5 to 7 minutes. Add miso mixture to saucepan and bring to simmer. Cook, stirring occasionally, until mixture thickens, 8 to 10 minutes. Cover and keep warm while noodles cook.
Bring 4 quarts water to boil in large pot. Add noodles and cook, stirring often, until almost tender (center should still be firm with slightly opaque dot), 3 to 5 minutes. Drain noodles and transfer to wide, shallow serving bowl.
Ladle sauce over center of noodles and sprinkle with cucumber, sprouts and dark green scallion parts. Toss well and serve.