Using a handheld slicer or mandoline, slice the vegetables as thinly as possible and no thicker than 1/16 to 1/8 inch. It's imperative that you slice the chips evenly so they cook at the same rate. Group the vegetables by type into separate bowls of ice water and set aside for 30 minutes.
In a deep fryer or heavy-bottomed stock pot, heat the oil to 300°F (150°C).
Prepare 2 baking sheets by lining them with multiple layers of paper towels or a brown paper bag cut open.
Drain and pat each group of vegetables completely dry using paper towels. (You can use clean kitchen towels if you’d like to save a tree, but you probably want to use paper towels on the beets as they stain something awful.)
Fry the vegetables in small batches, adding just a handful of chips at a time. (I suggest frying the beets last because they can stain the oil slightly and make all the chips a little rosy.) Fry the chips, stirring the vegetables gently using a long slotted spoon or stir-fry spider or chopstick to ensure even browning, until lightly golden and crisp, 2 to 5 minutes per batch. Try to maintain the temperature between 300°F (150°C) and no more than 325°F (170°C) so the chips don’t scorch. The exact cooking time will vary depending on the oil temperature and thickness of chips. When the chips are crisp, remove them from the fryer with tongs or a slotted spoon, allowing excess oil to drain away before placing the chips on the paper towels or bags. Salt the hot chips immediately and start the next batch of vegetables.
When you’ve fried all the chips, allow them to cool to room temperature (if you can manage to wait), toss them in a large bowl, and serve.