Use fresh soba noodles if there's a good Japanese market around, though there is absolutely no disadvantage to using dried, as we'll do here. All together we get a magical balance of acidity, richness and flavor with this unique dish.
Add soba noodles to a pot of salted boiling water and stir so it doesn't clump together. Meanwhile, make the sesame paste by placing sesame seeds, sugar and water in a blender and puree for 2 minutes. After 4 minutes, test noodles for doneness by trying one. Then drain in a strainer and run under cold water to remove excess starch and cool them down. Add a pinch of salt to the sesame paste and taste for smoothness. Chill in the refrigerator to cool down while the noodles are finished.
Place drained noodles in a large bowl. Using a mandolin, shave cucumbers on top of the noodles. Slice radishes in half, then thinly slice and add to noodles. Squeeze lime juice directly on top followed by a few large pinches of salt then, toss to combine. To serve, place a few spoonfuls of sesame paste in an individual serving bowl and spread all around the inside of the bowl. Place about 1 cup of noodles on top of sesame paste making sure the garnish is on top. Repeat to make 4 individual servings. Garnish with chives.