Line a fine-mesh strainer with 2 layers of cheesecloth and put strainer over a bowl. Put yogurt in strainer, cover with plastic wrap, and chill 30 minutes.
Tear romaine into bite-size pieces, rinse, and dry thoroughly between layers of paper towels. Arrange lettuce evenly in a 9- by 13-in. baking pan or other 2-qt. dish.
Chop sugar snap peas and arrange evenly on top of lettuce. Wash, trim, and thinly slice radishes. Arrange them on top of sugar snap peas.
Peel cucumber, halve lengthwise, and, using a spoon, scoop out and discard pulpy flesh and small seeds in the center. Cut each cucumber half lengthwise again and chop. Arrange cucumber on top of radishes. Trim green onions, thinly slice white and light green parts, and sprinkle evenly on top of cucumber.
If using frozen peas, put in a colander or strainer and run under hot water for 1 to 2 minutes to thaw. If using fresh peas, follow cooking instructions in Notes. Dry peas thoroughly on paper towels and sprinkle evenly over green onions.
Transfer yogurt to a bowl (discard liquid beneath strainer). Stir in oil, salt, and pepper, then stir in dill and mint. Spread yogurt mixture evenly over salad and sprinkle with chives. Cover with plastic wrap and chill for at least 2 hours and up to overnight. Serve cold, cut into 12 pieces.