Combine the peanut butter, soy sauce, vinegar, sesame oil, sugar, ginger, garlic, and cayenne in a blender and process until the mixture is smooth, about 30 seconds.

With the blender running, add the hot water one tablespoon at a time until the sauce reaches the consistency of heavy cream—it should be glossy and pourable but still thick enough to coat a spoon.

Bring 4 quarts of water to a boil in a large pot, then add the noodles and cook until tender, stirring frequently to prevent them from sticking together.

Drain the noodles in a colander and rinse thoroughly under cold running water until they are no longer warm to the touch. Drain them well, but leave them slightly wet so they stay slippery and do not seize up.

Transfer the chilled noodles to a large bowl and toss with the oil until evenly coated.

Add the radishes, scallions, cucumber, carrot, cilantro, and the prepared sauce to the bowl and toss until the noodles and vegetables are well combined.

Serve immediately in individual bowls, topped with a sprinkle of sesame seeds.