In a medium saucepan, heat the oil to 325 degrees (a deep-fry thermometer clipped to the side of the pan is helpful). You want the oil to be a few inches deep, with plenty of room for things to bubble up without spilling over as you drop in the chips.
While the oil is heating, slice the beets and radishes thinly with a mandoline or peeler.
Line a plate with several layers of paper towel.
Test the temperature of the oil by adding one slice of veggie to the pot. It should bubble aggressively and be golden brown in less than a minute. If this is the case, remove the test chip and flip it onto the paper towel to drain. Add a handful of chips, making sure to separate them as they go into the pot so they don't stick together. Turn them over as the edges brown, then take them out and drain them on the paper towel.
Continue in a similar fashion until all the chips are used up. Sprinkle them with a tiny bit of salt every few batches, but don't overdo it. Serve immediately.