Heat oven to 400°F. Wrap beet in foil, place on a small baking sheet and roast for 50 to 60 minutes until very tender; cool for 10 minutes, rub off the skin and cut into chunks.
Blend beet, chickpeas, tahini, lemon juice, garlic, oil, salt and measured cold water in a food processor for 2 to 3 minutes, scraping down, until smooth.
Toast the cut faces of the multigrain bagels under the broiler for 1 to 2 minutes and let cool.
Spread hummus evenly over both cut faces of all four bagels, using the full batch. Divide cucumber, radish and arugula among the bottoms in thin layers.
Close the sandwiches and cut in half with a serrated knife. Serve promptly, or keep the vegetables and hummus refrigerated separately and assemble close to eating.