Scrub carrots gently (so you don't lose the bright outer color), using a brush under running water. With a mandoline or vegetable peeler, cut thin lengthwise ribbons to make 1 qt.; discard ends and tough cores.
Put dark and light carrot ribbons in separate bowls of ice water and soak about 15 minutes to crisp. Drain, roll in kitchen towels, and pat dry.
Discard tough or thick stems from pea shoots and tear sprigs into 4- to 5-in. pieces. Pull strings from straight sides of snap peas, then thinly slice peas lengthwise. In a large bowl, whisk oil, lemon juice, garlic, 1/2 tsp. salt, and the pepper.
Add carrots, pea shoots, snap peas, feta, mint, and chicken to bowl and toss gently. Season with more salt if you like.
*Find at farmers' markets and Asian markets.