In a small bowl, combine the potato starch and 1/4 cup water and set aside.
In a large nonstick skillet, heat the vegetable oil over medium-high heat. Add the pork and cook, stirring often, until it is browned on all sides and some of its fat has rendered out, 6 to 7 minutes.
Add the onions, garlic, and ginger and cook, stirring occasionally, until the onions have softened, about 5 minutes. Add the potatoes, zucchini, and radish and cook, stirring occasionally, until the radish and potato start to turn translucent, 5 to 6 minutes.
Stir in the black bean paste, brown sugar, salt, and pepper to taste. Add 3 cups water and mix well. Simmer gently for 3 to 4 minutes. Stir the potato starch mixture, add it to the skillet, and mix well. Simmer, stirring frequently, until the vegetables are cooked through and the sauce has thickened, 10 to 12 minutes more.
Meanwhile, cook the noodles according to the package instructions (usually 5 to 7 minutes), drain, rinse well, and drain again. Divide the noodles among four to six large bowls. Spoon the vegetable mixture on top of the noodles and garnish with the cucumber, sesame oil, and sesame seeds.