In a 5- to 6-quart pan over high heat, bring 2 1/2 to 3 quarts water to a boil. Break pasta in half and drop into boiling water; stir to separate strands and cook, uncovered, until barely tender to bite, 5 to 6 minutes. Drain pasta, rinse with cold water, and drain well.
Meanwhile, rinse cucumber, zucchini, and apple. Trim off and discard ends of cucumber and zucchini; cut vegetables into 1/2-inch cubes. Stem and core apple; cut into 1/2-inch cubes. Add cucumber, zucchini, apple, and onion to cool pasta.
Pour any excess liquid off yogurt. In a bowl, mix yogurt, ginger, and 1/2 teaspoon salt. Add to rice mixture and stir gently to coat.
Set a 6- to 8-inch frying pan over medium-high heat; when hot, add oil and chili flakes and stir just until chili begins to darken slightly, 15 to 30 seconds. Add pepper. Pour over pasta mixture and stir gently to mix, adding more salt to taste. Spoon into a serving bowl and garnish with shredded carrots.