Preheat the oven to 400°F and line a baking sheet with parchment paper. Spread the diced beets on the parchment paper, drizzle with 1 tablespoon of oil, celery seeds, salt, and pepper and toss. Roast for 20 minutes, or until tender and easily pierced with a fork.
Prepare the soba noodles according to package directions, but slightly undercook them to al dente. Drain, rinse with plenty of cold water, and transfer to a mixing bowl.
In another mixing bowl, combine the scallions, cucumber, and white beans. Whisk the dressing ingredients together in a glass measuring cup or bowl, pour half over the bean and vegetable mixture, and toss. Add the remaining dressing to the soba noodles and toss.
Divide the soba noodles among serving bowls and twirl into a mound in the center of each bowl. Spoon the bean and vegetable mixture over the soba, garnish with roasted beets, and sprinkle with roasted walnuts.