Heat oven to 400°F. Wrap beet in foil on a small sheet; roast 50 to 60 minutes until very tender. Cool 10 minutes; rub off skin and cut into chunks.
Process beet, chickpeas, tahini, lemon juice, garlic, oil, salt, and measured cold water 2 to 3 minutes until smooth, scraping down.
Broil cut bagel faces 1 to 2 minutes; cool.
Spread full batch hummus evenly over both cut faces of all four bagels. Layer cucumber, radish, and arugula thinly on bottoms.
Close; halve with a serrated knife. Serve promptly or refrigerate vegetables and hummus separately, assembling near eating.