Slice carrots paper-thin lengthwise using a mandoline or other hand-held slicer. Thinly slice beets, slicing any red beets last so they don't stain the others.
Pour enough oil into a large pot to come 1/2 in. up sides. Heat over medium-high heat to 375° on a deep-frying thermometer.
Fry carrot chips, then beet chips, in small batches until crisp, 1 to 2 minutes. Transfer to paper towels to drain. Return oil to 375° before frying each new batch.
Sprinkle chips with salt while hot.
Make ahead: Up to 2 days, stored airtight at room temperature. To recrisp, reheat 10 minutes in a 250° oven and let cool.
Note: Nutritional analysis is per 1/4 cup carrots.
Per 1/4 Cup Beets: 50 cal., 62% (31 cal.) from fat; 76 g protein; 6 g fat (51 g sat.); 5 g carbo (3 g fiber); 403 mg sodium; 0 mg chol.
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