Preheat the oven to 375F. Put the carrots on a sheet of parchment paper. Sprinkle with salt, drizzle with oil, toss to coat evenly, and then wrap loosely in the parchment, sealing the edges. Bake until tender, 30-40 minutes. Open the parchment and continue to roast for 5 minutes. Let cool completely then transfer to a food processor. Add the tahini, brown sugar, cumin, and chile flakes, and process until smooth, scraping down the sides of the bowl. Add a little water if needed to make a thick dip. Transfer to a bowl and season with salt.
To make the chips, have ready 2 bowls of ice water. Using a mandoline, cut the turnip, parsnip, radishes, and carrot into paper-thin slices and add them to one bowl of ice water. Cut the beets in the same way and add them to the second bowl of water. Let stand 15 minutes.
Pat the vegetable slices dry on towels and arrange on a platter. Serve with the hummus.